I once had a debate with someone claiming science could get enough proof to disprove religion, and he refused to back his claim, even when the claim was about science capabilities and not religion itself, claiming the burden of proof was on me. Well, I was opposing his argument by claiming science can't gather enough proof, so you can see I had the negative stance.
Burden of proof is being abused, indeed. Also, it is a fallacy. If you limit to poke holes in the opposing argument, you will lose unless your contrary is really stupid.
But yet is being abused by dishonest debaters to avoid to show evidence or arguments to claims, as Rafe (up) says it. I think a debate must start SHOWING evidence or argument to support the main claims instead of starting a debate asking to the oponnent to show evidence using the "burden of proof".
Burden of Proof itself is a necessary component of debate or argument. A person making a positive claim has the onus to prove it. The person stating "BS- prove it" doesn't have a burden.
One of the most common fallacies used here (and anywhere really) is the shifting of the burden of proof, particularly in religious debate. One claiming that a god exists has made an extraordinary claim, and thus has a heavy onus. The person stating that god is unlikely or not supported by evidence doesn't NEED to prove anything, any more than I would need to disprove Sagan's dragon in my garage. The insistence by those using this fallacy that somehow, magically , people should be able to provide proof of universal nonexistence is just nonsensical and an obvious attempt to wriggle out of their own burden as they have no support for themselves.
The burden of proof is not a fallacy. It needs to be established in every debate so that those making claims can be distinguished from those who refute said claims. Making this distinction is by no means a lack of proof as it isn't even an argument; it is a necessary cornerstone of the debate process.
You must be able to back up an argument with some kind of evidence otherwise it's simply hearsay. So, the burden of proof is on the claimant not people refuting the claim. (e.G. If the law worked without the burden of proof resting on the prosecution there would be a lot more innocent people in jail)