Captain America doesn't have any secret identity, so removing his mask doesn't have much of any point. But while fighting Batroc the Leaper in Captain America: The Winter Soldier he specifically removed his mask to fight but why did he do that? Is it some kind of respect he is giving to a good fighter or is it for some other reason?
It's thematic for multiple reasons.
It's a continuation of Steve's character plot. Is Captain America just a symbol, or a man?
It ties into the overall plot of the movie. Who is the man behind the uniform? Contrast 'Who is the Winter Soldier' and 'Hydra infiltrating SHIELD'.
Batroc was Goading Steve into a fight. Or that is, in a no holds bar fight. Steve, finding that he doesn't need to pull his punches, follows suit. A common trope called The Gloves Come Off., as Steve changes mindset.
It's also a similar trope, The Coats Are Off, as Steve physically removes his protective gear. The Shield, as well as the Symbol of Captain America, his helm. Steve is no longer fighting as Captain America, but as Steve Rogers.
Batroc: I thought you were more than just a shield...
Rogers: Let's see.
The suit's theme is that Rogers is adapting to modern SHIELD. In First Avenger, he's a US Soldier, attached to Strategic Scientific Reserve, in a straight forward Good vs Evil fight. In Winter Soldier, he's a spy/black ops agent, in a fight where he doesn't know where he or SHIELD really stands. Then the fight with Batroc, it's very straight forward. Just a one on one fight. He removes the helm to unrestrict himself. He's detaching himself from the new SHIELD, the new world, and going back to himself.
GeekTyrant.com: Ok, I have one really quick question before I go about Cap’s new suit, just because my husband is a designer and he loves it. So what went into developing it and what changes did you think were necessary?
AR: We put a massive amount of time and energy into that suit, because it was so important to us. I mean, look, for us it was Cap in this movie, we wanted him to start in a place of trying to embrace the modern world in this film, and not sort of being stuck in the past, even though he is, inevitably. So we wanted him to be training himself in the couple years he’s been alive in the modern world in all the most current fighting techniques, all the current strategies and military moves, et cetera. And we wanted him to be able to sort of craft, he wanted his suit to be an expression of the new soldier he was trying to be. So that suit is very much drawn upon the most current thinking and technology in terms of military war apparel. And that’s basically what that expression is, more or less. And he starts the movie in a sort of a night mission, we call that the stealth suit, we referred to it for a long time as the stealth suit.
AR: Yeah, but and also, the suit is very thematic in the film as well in the sense that like he begins the movie in this suit that’s very S.H.I.E.L.D., you know? His old suit used to be a symbol. He’s like no longer… the emphasis is off the symbol in the new suit and more on the functionality. And of course, as the movie goes along he realizes he needs that symbol again.
This is further explained thematically, as Rogers doesn't really trust the new SHIELD. The following three scenes after the fight involve him, arguing with Natasha, then Fury about the secrecy and spy bullshit, the lack of Steve's army upbringing, then Steve going back to his roots (via the Smithsonian exhibit), and finally talking to old Peggy about his doubts. It cements how the fight, without the new SHIELD uniform vis a vis the helmet, reflects Steve's outlook on SHIELD.
Is it some kind of respect he is giving to a good fighter?
Yes, in a way. It's also Steve telling Batroc, "you have my full undivided attention." In the comics, Batroc is a long time foe to Captain America, an almost equal (when he's not shown as a joke character). He's not a random cannon fodder grunt. He holds his own against Capt, Wolverine, Spider-man, etc.
Side Note: Steve normally has to hold back, as he's
superhuman, and normally can't fight at full strength. Batroc shows him that he doesn't need to hold back.
I think it's part of the cameo from George St-Pierre. He's having a fight vs a UFC champion, so to complement the cameo he goes without a mask to make it feel like a UFC fight. I have not found anything from the director specifying an exact reason so I assume it's related to GSP's UFC carreer.
As OP stated, he doesn't need to hide his identity, it's purely for the show, just like in an ufc arena a masked fighter would arrive with his mask to finaly remove it before the event.