It could be just a flattering color to dress them in in TVD, but I don't think so: it starts popping up ALL THE TIME once Caroline and Tyler form their friendship, although neither of them are really dressed in it before. Caroline wears red in 1x03 (when she is directing cheerleader practice) and Tyler wears it only in his football scenes (where he mostly gets yelled at for doing the wrong thing). I think that there is a pattern of agency/support when they are dressed in red, and their relationship gives them extra motivation to act/support. All screencaps come from homeofthenutty
Even back when Tyler and Caroline weren't even friends, red appears.
From 1x07 (Haunted)
Caroline: I'm going to drink until someone is hot enough to make out with.
Tyler: Sounds like a plan.
This is the third time they are in a scene together, and the first time they actually talk to one another onscreen. And he's totally supporting her choice here! LOL Awww look at them thinking the same thing and not even considering each other. Might be coincidence, but could also be foreshadowing.
And then there is Tyler's "We people" speech in 1x10 (The Turning Point) which starts to really underscore that foreshadowing. Notice what color Tyler uses as his example?
Tyler: So what's up with you and Forbes?
Matt: Nothing's up.
Tyler: I saw you two in the hall today. Don't even try to deny it, bro. You're tapping that.
Matt: It's not like that.
Tyler: Never is. Until you become "we" people.
Matt: "We" people?
Tyler: Yeah. "We can't make it to the party." "We'll never miss a game." "We don't like the color red."
Matt: We hung out, like, twice.
Tyler: Like I said. "We."
In retrospect, the irony here is great- if anyone ends up "we people", it's him and Caroline as a couple. But in Season 1 and at the start of Season 2, they may be in the same social circles, but are just friends of friends.
From 2x07 (Masquerade)
This is the kickoff for the start of the true connection between them- and oh look: there's Caroline in a red dress.
Caroline has so much agency in this episode- she finally sticks it to Katherine by luring her into a trap. And right after that triumph, Caroline goes one step further when she realizes what is happening with Tyler- she steps in to protect him first from Matt, shouts a warning when she sees Sarah with the knife and then protects him from taking blame for Sarah's death.
Tyler: What are you doing?
Caroline: I'm fixing...a very bad situation.
Tyler: I did this. I killed her.
Caroline: No. You didn't mean to. And I think it's best for everyone-
Tyler: But she's dead. You don't know what that means.
Caroline (looking regretful): Actually...Tyler, I think I do.
Tyler (shaking his head): No you don't, Caroline.
Caroline: ...Has your wound healed?
Tyler (opens up his shirt and checks before looking at her, astonished): How did you...?
For someone who has been so powerless in the series up to this point, this is huge. And she doesn't step up just for herself in this episode, she steps up for someone who she's never been close to. And the friendship that sparks is something that will continue to be a source of strength for her throughout the series.
From 2x11 (By the Light of the Moon)
Caroline: I want to help but I don't know what to do.
Tyler: There's nothing you can do.
(His bones start breaking again and he screams. She muffles her crying.)
Tyler: Get out!
Tyler: Get out! I don't want to hurt you!
Caroline: No! No.
(She gets down and holds him.)
Watching Caroline comfort Tyler through his first brutal transformation is so heart wrenching. What's even more amazing is knowing she's doing this despite the risk to herself- she knows he could kill her but she just keeps refusing to leave him to face this on his own. And of course, she's wearing red- a red shirt under a red leather jacket. As Tyler says later- "she's pretty incredible." He recognizes just how brave and caring she is for choosing to help him.
From 2x12 (The Descent)
Caroline: Why is it so hard to let someone help you?
Tyler: That's not it-
(She whirls back to face him, seeming to get a little annoyed.)
Caroline: Yeah it is, Tyler. It's like you don't want anyone to care about you and I'm sorry- I care. I care, Tyler. So forgive me if I overstepped my bounds-
(He kisses her. He pulls back and then they kiss again, more passionately, until she suddenly pulls back and brings her hands over her mouth.)
Caroline: You can't do that.
This time he's wearing the red, and he's making the first move. rawkry has already made a great post about this scene in particular, but I just wanted to say something else about this episode: Tyler thanks Caroline for her help, also while wearing his red football jersey earlier in the day. And I'm pretty sure that is the first time we ever see anyone thanking Caroline for helping them in the series, even if they have asked her to do something for them. Hell, she often gets disparaged and insulted instead (Damon, Kelly Donovan, Matt), or that help is merely taken for granted (Elena, Stefan)- how utterly refreshing that somebody actually appreciates her efforts.
From 2x16 (The House Guest)
ETA this one- Caroline chooses to act on a lot of things here- she tells Matt how she feels, she takes on Dr. Martin to protect the others, and then she feeds Matt her blood to save his life and tells him the truth about herself. Even her fingernail polish is red in this episode.
From 2x20 and 2x21 (The Last Day and The Sun also Rises)
Matt: Caroline! The door's not going to hold!
Caroline: Come on! Tyler, please!
(Matt shoots wolf!Tyler.)
Caroline: Matt! No! Hey, no! It's Tyler!
Matt: He's trying to kill us!
(wolf!Tyler is lying on the floor, wounded. Caroline starts unwinding the chains on the gate.)
Matt: Caroline, stay back!
Caroline: We can go around him.
Matt: Are you nuts?
(Caroline shakes her head.)
Caroline: You're not going to shoot him again, OK? Matt, take my hand.
Caroline wears the same outfit for these two episodes- a red shirt under a white jacket. (Caroline and Tyler are also represented by Light/Dark, so this outfit is doubly symbolic). Again she's the one taking action- she confronts him about leaving, does her utmost to protect Tyler from Matt (and Matt from him), and finishes off by telling Tyler what to do and even what to say. LOL
From 3x02 (The Hybrid)
Tyler is dressed in red again, in the first of three episodes where he is the one supporting Caroline. He defends her to his mom first thing in the morning ("And Caroline...not a prostitute.") the way he was unable to defend Vicki in season 1 when Carol refers to her as "trash". Awww, look at him grow. :) But the biggest thing he does is this:
Tyler: You think Caroline's a monster? I'm the monster, mom.
He comes out to his mom about being a werewolf in order to change her mind about Caroline, turning something he has zero control about into leverage to protect her. This episode also makes me all misty now that we know Carol's fate- she has always accepted him unconditionally, always tried to protect him and that is not something every parent can do.
From 3x03 (The End of the Affair)
Tyler's wearing red again- he's the one who goes to Liz to find Caroline and break her out, but again, his role is mostly supportive even though he's taking action.
From 3x04 (Disturbing Behavior)
There have been a crap-tonne of posts on this particular scene recently so I'm just going to point out the red shirt and supportive role Tyler plays here and you can click on the links for others' thinky thoughts on the whole thing. This is my absolute favorite Forwood scene. <3
From 3x06 (Smells like Teen Spirit) Tyler chooses to exercise his brand new ability to compel humans here by forcing the coach to end practice early so the team can go to the bonfire and get drunk. Caroline is unimpressed by his behavior.
From 3x09 (Homecoming)
Caroline: But...you don't have any true control over yourself.
Tyler: I never did before. The full moon controlled me. After everything we've been through- you've been there for me through all of it. Don't turn your back on me now.
Caroline: (hesitates) Tyler...
(He looks like he's on the brink of tears.)
Caroline: I just...
Tyler: I got it.
(He leaves abruptly and she looks devastated.)
Caroline wears red when they break up here. And she doesn't even say the words, he gets up and leaves before she can, so I think this falls into limited agency again, although her intent is clear.
From 3x12 (The Ties that Bind)
Red appears twice in this episode, both times on Tyler. First as a belt when he tries to turn to break the sire bond:
And then as a red t-shirt (under a black coat) in his last scene with Bill Forbes, who tells him the turning process has to be painless before he can truly be free of the sire bond. We find out in 3x14 that "[he] just kind of kind of freaked out and left. ...[He's] out doing what [Bill] said. [He's] going to fix [him]self and come home to [her]."
From 3x20 (Do not Go Gentle) They both wear red! He just has it on his tie, she has a whole outfit. He's choosing to be at the dance because he doesn't want to "hide while Klaus is macking all over [her]". And she finally tells him she loves him- something he'd said to her twice already.
From 3x21 (Before Sunset) This is the scene where they first exchange "I love you"s over the phone. Caroline ends up mostly being a victim the rest of the episode, but she does choose to do a couple things: comfort Rebekah over the loss of her mother and then save her life when dark!Alaric goes to stake her. I'm still wondering if Rebekah will ever pay her back.
What puts a real dent in my agency/support=red theory is that Tyler doesn't wear anything red when he defies Klaus to save Elena in this episode. He's even in the supporting role restraining Klaus while Stefan's doing the actual blood connection thing while they're dessicating him. But you know who else is wearing red in this episode? Bonnie, Abby and Alaric. So maybe it's not just a Forwood thing? This may totally be my shipper goggles restricting my view here.
Caroline obviously goes home and changes and they wear these outfits into the next episode- Caroline makes everybody celebrate putting down Klaus and "smothers" Elena a bit (and Elena actually appreciates it). But she doesn't really make any big choices/actions until the next episode...
...where she agrees to leave with Tyler when he asks her to run with him. (3x22 The Departed.) Caroline is very much a part of the save-Elena-no-matter-what group (although what exactly she is running off to do in this episode is never explained- good job there, writers) so having her choose him and herself (and their moms) over them is HUGE for her.
She's still wearing the outfit when she thinks he's dying right in front of her. And she very reluctantly leaves as he asks when he makes her, which I thought was weird- he's already been completely vulnerable in front of her during his first change, and they weren't even as close then. ...And then we find out it was Klaus the whole time, and that Tyler has given up his body in order to save everybody. And he still isn't wearing red.
From 4x01 and 4x02 (Growing Pains and Memorial)
Caroline: Stop! You're killing him!
Caroline struggles to get Tyler back in his body for most of Growing Pains. Without Caroline's pleas I don't know that Bonnie would have caved- she is pushed around and finally loses so much in this episode and I don't think that anybody in the group even really noticed except Jeremy.
and then the same outfit while she watches Stefan pull the bullets out of him in Memorial. Mostly just emotional support here, although I bet she was the one who told Tyler to go to Stefan for help- doesn't seem like what his first choice would be.
4x03 The Rager
While Caroline is wearing red in this episode, she actually doesn't show the most agency/support with Tyler for once- he is actively pushing her away in this episode. However, you see her giving her support and help to STEFAN when he asks.
Caroline: Come to me. Whenever you want. I won't let you lose control.
From 4x06 (We all go a little mad sometimes) First red outfit is for their "fake break-up" scene, where Caroline returns Tyler's things, including his team jersey (also red) to start off the ruse. Caroline is supposed to be a bad liar (1x08, 2x04, 3x11), but she's actually convincing when she's lying to help in a plan she believes in (2x07, 3x19, every "Klaus bait" scene.)
Caroline changes back into red later when she reveals that "[she] agreed to go on a date with [Klaus] to help keep up the ruse between [him] and Hayley." She's trying to be supportive of Tyler's goals- but it hurts him, as had her earlier betrayal of Chris, whom he had promised to protect.
4x08 We'll always have Bourbon Street
Tyler is back in red for the whole episode, but I would argue that despite the alpha status he gains by the end, he still isn't in complete control of what happened here: Hayley is the one who pushed him to become alpha, not only at the Grill but also on the phone before they rescue Caroline. But he does make the decision to force Kim to submit.
Neither of them are dressed in red in 4x09 either despite a lot of decision making/acting/supporting, but I think it's because the directors went light/dark contrast instead. Notice who is dressed in red and her final act? I so wanted to hate her too. LOL
Final verdict: it isn't a direct correlation, but I think there is a connection between when Tyler and Caroline are dressed in red and their actions onscreen. For two characters who have very little power in the Mystic Falls circle, I think it is telling when they choose to act on their own rather than follow the lead of somebody else.
Tyler makes it clear that Caroline is his motivation in season 3. When Bill asks him how bad he wants to break the sire bond in 3x12, he looks right at her- and then starts to force himself to turn. He outright says, "by breaking every bone in my body a hundred times for the girl I love" in 3x21. It's nice to see that he's moved past doing everything for her in season 4, taking one for the team in 4x02 (not wearing red, but come on- it was a funeral) and then taking on the un-siring project for both himself (revenge) and to help the hybrids in his pack get free for the rest of the season thus far. But I'm still waiting for Caroline to really choose Tyler and herself over her other main motivation- the save-Elena-group. Because despite what she says, ("Tyler, I just got you back. I'm not losing you again." 3x19) she doesn't plan t do anything to help him until 4x09.
They still act mostly supportively when they choose to act ("You help your friend. That's what you do." 2x13). Which is funny when compared with the "Queen Bee" type that Caroline is supposed to be, but all her club/dance committee/student council activities don't impact her social currency- she is never the director of operations for their group until 4x09. Rather, she is aware of how much she owes Stefan and Elena for "sav[ing her] vampire life" (4x03) despite the fact that she has directly helped to save him twice in season 2, and despite Elena forcing her to come out of the vampire closet to her mom the first time.
The red=action probably extends to the other characters as well, but I tend to rewatch mostly T/C scenes, so I know this isn't a very complete analysis. Sorry. :P But I know I'll be watching for red, especially in that final showdown against Klaus, which I praaaay is coming. Please