Why should theists prove that God exists?
When atheists and theists debate the existence of gods, atheists generally ask for some sort of proof of the god or gods that the theist claims exist. Why do they do this? That is a difficult question to answer because while it is true that atheists often ask theists for proof that their god exists, sometimes they shouldn't. Asking for proof may end up confusing the important issues, causing distraction, and leading the conversation away from where it should be.
If that is the case, why do some atheists always ask for proof? I think that there are two reasons one a question of semantics and one a question of principle. First, however, it would be easier to explain what atheists should be doing and why.
If a theist claims that a god exists, an atheist is justified in asking for that claim to be supported. Support is not the same as proof, although proof is a type of support (a very good type, in fact). Claims need to be supported if they are to be taken seriously. Anyone can claim anything at all, and a principle means of distinguishing all the nonsense in the world from the reasonable ideas is the latter have good support and the better the support, the more justified the claim is.
Thus, when you are asked to support a claim you have made (no matter what the subject this isn't limited to debates over the existence of gods), you are being asked to show good reasons why your claim should be taken seriously and perhaps even believed. There are lots of claims out there, most of which probably aren't worth believing, and you can't expect others to necessarily give your claim more attention or respect than all the others they encounter on a daily basis.
Presumably you care about that at least a little bit, or else you wouldn't have made your claim in the first place. Thus, by making a claim, you have essentially taken on an intellectual and a moral obligation to offer some support for it. Unless you believe things randomly, then you must already have reasons for belief that you consider good; they, then, should be the first ideas you offer as support.
The reason why one should ask for support rather than proof has to do with the way these two terms are used. Typically, to say that a claim has been "proven" implies that it has been demonstrated as true to such a degree that dissent and disagreement are no longer reasonable or even possible. It has a feeling of being "absolute" and permanent. As a result, "proof" is treated as being a very high standard to meet and it is no surprise if someone might object to being asked to meet that standard.
No one, however, can reasonably object to being asked to support their claims. If they think their claim is rational, reasonable, or justified, then they must think they have rational or reasonable support for their claim which justifies believing it. How can they not be asked to provide it when they put their claim out in public? Being expected to support one's claim is a standard which applies to everyone who makes an empirical claim.
So why do atheists ask for proof? As stated above, one reason is semantic it seems that, in some cases, it is simply an error in word choice. The atheist isn't really asking for an absolute, final and irrefutable case which demonstrates the existence of god, but is instead asking for good grounds for belief. The word "proof" is simply one which is commonly used and which comes out naturally. Debates about the existence of gods certainly isn't the only context where someone might use the term "proof" when they really mean "support."
Other times, however, the atheist really is asking for "proof," and this is more serious. It is, as I have indicated above, often an error. We don't normally ask for such absolute proof for claims on other topics, and while it might be nice to have such proof of the existence of a god, it shouldn't necessarily be demanded here. It also isn't really vital because, in my experience, theists aren't able to provide any good support for belief in a god, so why bother placing the bar higher and push the conversation off onto a tangent about the nature of "proof"?
There is one exception to this principle: asking for proof of a god is reasonable and justified if and when a theist suggests or even states outright that they have such proof. Sometimes this statement may be explicit and they'll use the word "proof," but other times it will implicit and they'll use words that merely imply that they have proof words like "definite" or "undeniable." It isn't too common for a theist to go in this far, but when it does happen, then requesting that proof not only makes sense, but it's the right thing to do.