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A few weeks ago, a student asked me during my office hours whether I was religious or not. More specifically, he asked whether I believed in his religion, and was visibly disappointed when I replied negatively.

How should I react to a question such as this? Religion is a bit of a touchy subject, and even if I am a TA (i.e., not the instructor of the course - and more importantly, also a student), I don't want to be put in a similar situation again.

More generally speaking, how do I react to a situation in which I have to answer a question in a way that I know is likely to upset a student, without lying or making the situation worse?

You address this by stating directly

"I would prefer not to discuss this topic during office hours. Can I help you with any questions you have on the homework?"

If the issue persists, I would speak with the professor and perhaps also your school's diversity office (or something similar).

It is like a question about your political opinion or sexual preferences: It is your private matter!

If you want to tell the student, you can do so, but you'll have to face the discussions (which can be fruitful or stressful), but if you prefer not to share them (which many people will do in a professional context), tell the that this touches your private field and since this does not belong to the workplace, you do not want to answer this question.

In the US, you have done just the right thing. Tell him the truth and leave it at that. You can't avoid such situations as they are set up by others.

Of course, you can say, and it is perfectly valid, that such questions are very personal to you and you don't feel that you want to discuss them. Reasonable people will accept that, and if they aren't reasonable, they have no right to proselytize. You don't need to be harsh ("None of your business"), but you have no obligation to answer.

There are countries, of course, that have State Sponsored Religion in which the only accepted (safe) answer is the State Religion, but not here.

and was visibly disappointed when I replied negatively

This is their problem, not yours.

How should I react to a question such as this?

If you feel uncomfortable defending your position when it comes to religion, politics or sex, or simply you don't want to discuss them with an extraneous person, cut it short and answer that you're there to just answer questions about the subject you TA. And, anyway, you don't have any obligation to answer questions like that.