Manbearcat
Legend
That's not a no-win situation. That's the DM saying that all orcs are irredeemably evil, and that the duty of the paladin in his milieu is to destroy all orcs. In that situation there is a win - kill the orcs.
Well. Its not a "no-win situation" if your agenda is to play skillfully as a player by internalizing the GM's conception of your character's ethos requirements and abiding by them despite any conscientious objections you may personally have (either as a player or vis a vis your own conception of your paladin's sense of honor, sacrifice, duty, etc).
If your agenda is to play out your own conception of your paladin (be it classically Romantic virtues or another specified code of virtues) by being thrust into and tested by thematic conflict, to find out in play how your conceived paladin prioritizes the tenets of this ethos system when its under fire, then it is 100 % a no-win situation for your (sincere) conception of your paladin to be overturned by the veto power of a GM. This, of course, presumes sincerity on behalf of the player.
I've never been a player and I've never had anything like the above arise at one of my tables. But I cannot imagine being a player of a Paladin and being told by a GM "yeah you must kill these defenseless, but rotten to the core (!), monstrous humanoid infants...or your God hates you...I was just talking to him and he said he would really, really, really hate you if you didn't kill them...nothing earns his ire more than his holy warriors ignoring their duty to eviscerate the babies of evil monstrous humanoids...so do you kill them or are you a chump of a fail-player and cast out of your God's good graces?"
"Well, I used to think that my Paladin was a defender of the meek, the innocent, and those who cannot defend themselves. I used to think that he would scream "DEATH FIRST" with divine power when facing a group of depraved townsfolk with torches and pitchforks who demand he stand down so they can put the orc baby he carries in his arms to the sword. But I guess that is just rubbish. Yeah, ok, I kill the orc baby and win. Bro-fist to my God."
I can't believe this sort of (comedic silliness removed) stuff actually happens. But if it does, it is indeed a no-win situation for a sincere player whose conception of their PC is utterly subordinated by their GM.