Is there a word/term for a question where the asker knows he'll criticise any answer?

I upvoted David's loaded question because it's a very common usage, but on reflection I realised that's not quite right for OP's context.

A loaded question is nearly always one that's asked in such a way as to force or encourage a particular answer (that the answerer might not give if the question were presented "fairly").

But a trick question is one where the questioner usually doesn't care what you answer - you'll be wrong no matter what you say. I don't normally cite Urban Dictionary, but here's their definition...

An inquiry having no correct answer, or one asked for the sole purpose of starting controversy or eliciting certain responses. Basically, a no-win situation.

Girlfriend asks: "Do I look fat in this?" (trick question)

If you tell her she does, she'll throw a fit and tell you how insensitive you are. If you tell her she doesn't she'll call you a liar and go off on a tangent about how "all men are the same" or some nonsense like that.

My favourite trick question is "Have you stopped beating your wife yet?", but David's "When polar bears eat penguins, do they get indigestion?" is neat too. But they're both slightly different to the example cited above (that has two wrong answers, ours don't really have "answers" at all).

I would call that a "loaded question."

A loaded question is one where the person asking it has an agenda behind it. While there are other cases where a loaded question is the appropriate term, I believe this to be one type.

Of course, one can say that traditionally a loaded question has some information that forces the other person to agree to unsavory terms to answer the question. See here

But, I would still characterize this as a subtype of loaded question.

Essentially the asker of this type of question is only asking it as a means of embarrassing the other person. Hence, my characterization.

It does not give any name to this type of question, but there is an excellent discussion of the usage in James Joyce's Portrait of the Artist as a Young Man.

Then he went away from the door and Wells came over to Stephen and said:

— Tell us, Dedalus, do you kiss your mother before you go to bed?

Stephen answered: — I do.

Wells turned to the other fellows and said:

— O, I say, here’s a fellow says he kisses his mother every night before he goes to bed. The other fellows stopped their game and turned round, laughing. Stephen blushed under their eyes and said: — I do not.

Wells said: — O, I say, here’s a fellow says he doesn’t kiss his mother before he goes to bed.

They all laughed again. Stephen tried to laugh with them. He felt his whole body hot and confused in a moment. What was the right answer to the question? He had given two and still Wells laughed. But Wells must know the right answer for he was in third of grammar.