You might try, “trick question” or “hostile questioning.” Hostile is more about the tone or intent; a trick question is more about cleverly posing a question with no good answer.
Other words for unpleasant, relentless questioning are badgering, haranguing, berating or hectoring—although those all connote rudeness and anger, while a trick question is more often outwardly polite.
Tommy Tran’s suggestion, “loaded question,” and Pam’s suggestion, “leading question,” are both good. “Leading question” is more likely to be used in a courtroom, but it also has a more specific meaning as legal jargon than in everyday speech. If I complain, “That’s a loaded question,” I’m accusing the other person of trying to imply something he or she is not willing to say directly. If I complain, “That’s a leading question,” I’m accusing the other person of trying to make me say what he or she wants, or put words in my mouth. Neither refer only to trying to make the person being questioned look bad; a lawyer coaching me to sound good or stick to my story might be told to stop “leading the witness,” or a question might be “loaded” with the implication that someone else is bad or wonderful.
Another related legal term is “complex question,” although unlike the previous two, this isn’t a term in common use. It means that a question presupposes something false. The classic example is, “Have you stopped beating your wife?” (Especially in a context where a witness is only allowed to answer “Yes” or “No.”) This has an even more obscure Latin name.
The word “abuse,” as in “abusive question” or “abusing the witness,” is the closest piece of legal jargon but has become obsolete; the word means something more violent today.
Another related legal term is “perjury trap,” where a prosecutor or investigator asks a suspect about some wrongdoing and you’re in trouble no matter what you say. If you confess, you’re guilty. If you deny it and they think you’re lying, they can indict you for that. If you refuse to answer on the grounds that it may incriminate you, that makes you look suspicious.
As a literary example, you might look at Mark 12:13. A few traditionalists familiar with the King James Version might recognize the obsolete turn of phrase, “to catch him in his words,” but modern translations say something like, “to trap him in a statement” or “to trap him by what he said.” This meant very politely asking for a yes-or-no answer to a question that sounded reasonable on its face, but where any straight answer would get you in trouble with somebody.
Mathematics has “coffin” or “killer” (in the past, when they were used to keep Jewish students out, sometimes called “Jewish”) questions:
The Mathematics Department of Moscow State University, the most prestigious mathematics school in Russia, had at that time been actively trying to keep Jewish students (and other "undesirables") from enrolling in the department. One of the methods they used for doing this was to give the unwanted students a different set of problems on their oral exam. These problems were carefully designed to have elementary solutions (so that the Department could avoid scandals) that were nearly impossible to find. Any student who failed to answer could be easily rejected, so this system was an effective method of controlling admissions. These kinds of math problems were informally referred to as "coffins". "Coffins" is the literal translation from Russian; in English these problems are sometimes called "killer" problems.