I have had someone to whom I was very close, in my past, tell me they were going to commit suicide if I did not do x. I did not get all worked up about it. I didn't do it, they did not kill themselves and we are not in touch much because I failed at caring enough (their words). I am fairly immune to that sort of thing because my mother did it at me all the time (not suicide - but she was going to leave the family because I did or did not do X, and other things of the sort. I saw my sibling react with hysteria in order to get her to stop and I would get accused by both of not caring or driving her away or other things. I did care but she never followed through and it felt like she was trying to steal my soul - I did feel manipulated by her antics because she was trying to get a specific reaction out of me. And I resolved to not be manipulated by others in that fashion - and I am not). I guess what I am saying is that I did not respond by feeling trapped by the person nor responsible in any way for what they did or did not do. I didn't want them to be dead, and I would have missed them, but not my problem if they chose to die.
I do however think of it as different in the case of a therapist than with real people.