One sense of "burden of proof" seems to be a game-rule for a (non-Bayesian) adversarial debate game. It is intended to exclude arguments from ignorance, which if permitted would stall the game. The players are adversaries, not co-investigators. The player making a novel claim bears the burden of proof — rather than a person criticizing that claim — so that the players actually have to bring points to bear. Consider:

A: God loves frogs. They are, above all other animals, sacred to him.
B: I don't believe it.
A: But you can't prove that frogs aren't sacred!
B: Well of course not, it never occurred to me to consider as a possibility.

At this point the game would be stalled at zero points.

The burden-of-proof rule forbids A's last move. Since A started the game by making a positive claim — the special status of frogs — A has to provide some evidence for this claim. B can then rebut this evidence, and A can present new evidence, and then we have a game going:

A: God loves frogs. They are, above all other animals, sacred to him.
B: I don't believe it.
A: Well, the God Book says that God loves frogs.
B: But the God Book also says that chickens are a kind of flea, and modern taxonomy shows that's wrong. So the God Book isn't good evidence.
A: I found a frog once that had the word "God" encoded in the spots on its back in Morse code.
B: But the spots on frogs' backs are probably pretty random. How many frogs did you have to check?
A: Umm ... a few thousand. It was a sacred duty!
B: But it would be a lot more convincing if all frogs had that pattern, wouldn't it?
A: Well ... Frogs are sacred in Homestuck, which is the most financially successful webcomic of all time. Surely that's a sign of God's favor.
B: They're sacred to Prospitians, yes, but Dersites think they're blasphemous. Besides, if financial success was a sign of God's favor, we should all be worshiping Berkshire Hathaway, not frogs.

According to the rules of the game, B doesn't have to establish that God hates frogs. B just has to knock down each one of A's arguments. Then, since A has failed to establish any evidence that holds up, B is (so far) winning the game.