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 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.
This goes into levels of stupid in Dragon Ball Z Abridged. During the second half of Goku's fight with Cell, everyone notices Goku's starting to lose steam. Trunks angrily tells the other Z Warriors to stop going by the tournament rules and help him. Vegeta tells Trunks off, claiming that his Saiyan heritage would refuse such help... only for Goku to declare he gives up. When Gohan is tagged in, he believes he can take the weakened Cell... until Goku tosses Cell a Senzu Bean. This earns him a What the Hell, Hero? from everyone.
Hellsing Ultimate Abridged: Walter's betrayal is reworked into this. He'd rather fulfill his duty as a vampire hunter and destroy Alucard, than his fulfill his duties to Hellsing (and Integra) and to a currently-being-overrun-by-vampire-Nazis England.