Who has the burden of proof when trying to prove or disprove someones religious beliefs? I have always believed it belongs to the person who is making a claim but can there be exceptions?
The way you have worded this is rather weird, so let me explain it in another way. In the simplest sense, the burden of proof lies on the mind who is making the positive claim. If someone is claiming the existence of a god or deity, or a multitude of them, or of anything supernatural or extraordinary, they have a rather large burden of proof. Few, if any, philosophers will state that it is possible to prove a negative (i.e. disprove the existence of a god).
The way in which you worded the question is such that it seems you are referring to someone who is trying to provide evidence that a certain deity does not exist. In that case, it would be the job of the person making the claim that the deity in question does not exist, which most philosophers will agree is impossible. However there is no sense in doing so unless they have already met their required burden of proof.
When in a debate, it is always the job of the theist to provide evidence for their deity (except presuppositionalism which is wholly dishonest, fallacious, and ridiculous).
Ockhams Razor is a philosophical tool that states that among competing hypotheses, the one with the fewest assumptions should be selected. Other, more complicated solutions may ultimately prove correct, but—in the absence of certainty—the fewer assumptions that are made, the better. - Wikipedia
Agnostics and skeptics will argue that there is no certainty re. the existence of God, and since postulating God's existence involves making more assumptions it is better to assume He does not exist. In areas where there is uncertainty, the burden of proof lies with those offering the more complex answer.
A theist could retort, firstly, that certainty re. God's existence is attainable, or second, that postulating God's existence does not require adopting more assumptions, or third, could reject Ockam's razor as a valid philosophical tool. In either case, the burden of proof would lie equally with both parties, each needing to defend their respective positions.
Generally in a debate setting the person who holds the affirmative position is laden with the burden of proof and at the same time is given the chance to go first as to offset this burden by giving him / her the chance to open the debate and set the tone.
In regards to a religious debate generally the theist is laden with the burden of proof but this is more a by product of the bias in the debate topic than something that is automatically put on a theist just by the mere fact of him being a theist.
With a simple change of the topic the atheist can be laden with the burden of proof and also given the chance to go first. Just imagine the topic... "Is atheism true?" or "Is atheism a justifiable world view?"