Neil Sinclair of Survival of the Fittest fits this trope. The primary example of such behaviour is trusting Dominica Sharpiro by offering her a place in his Pro escape group, despite knowing, for certain that she earlier killed another group member who became separated from the others.
The Nostalgia Critic is big on this one, protecting friends and children to the most extreme degree even though he knows full well it'll get himself hurt.
In Worm, the trope is discussed in Snare 13.10 when Grue i.e. Brian is talking to Taylor i.e. Skitter:
Brian: I worry about you. You throw yourself into these situations like you don't care if you die, like you've got nothing to stick around for except for those people you insist on protecting. Dinah, the people from your territory. People you barely know, if at all. And then you actually make it out okay, so you do it again, only more so. Riskier stuff. I start thinking about how I'm supposed to protect you, get you to stop, get you to focus on a goal that's actually attainable, because you're so capable that you could be amazing if you stopped acting suicidal.
In Red vs. Blue, Sarge refuses to use sniper rifles and other long distance weapons other than regular guns, as he believes the only way to kill someone is up close and personal. He admits that he has no problem with using a nuke on an enemy because of Rule of Cool.
In Sherwood Forest, Will follows Robin to the castle and rushes into battle to save him. This would be very noble if not for the fact that he's a terrible swordfighter and basically just manages to kill a guy through sheer dumb luck. He also got lucky in that his appearance made the Sheriff's guards scatter; if they'd stuck around long enough to realize he was just flailing wildly with a sword, they probably would have killed him.
Tempered Steel from Fallout Is Dragons will always, always, always try to talk others into doing the right thing. Even a pissed off dragon who is currently preparing its breath weapon.
Actually Invoked in Death Battle as an argument against Goku in "Goku vs Superman". Many Dragon Ball Z characters, Goku included, have a habit of demanding a fair fight even against an obviously superior opponent. Hence, even if Goku managed to figure out Superman's weaknesses to kryptonite and exposure to a red star, he would refuse to exploit them.