If you make a claim you need to have evidence to back it up, otherwise you shouldn't expect to be taken seriously. You can't just say 'well the opposite hasn't been proven yet so I'm going to believe it'. Just because it's more believable to say that a chimp doesn't live in my fridge, I can't claim that one doesn't in a debate scenario unless I am prepared to give evidence. If nobody explicitly claims the positive, then there is no way of shirking the responsibility of evidence onto someone else. If nobody claims that there is a chimp in my fridge, but I claim that there isn't, why should other people prove the claim they didn't make? It's my responsibility to prove the claim I made.

In your chimp scenario, why would anyone believe that a chimp is in the fridge without evidence for it? Is it relevant to disprove something that hasn't been proven? No. It's unnecessary and impossible in an infinite space.

That doesn't stop it being necessary in a debate scenario. If you can't prove your claim, don't make the claim. It may very well be the case than during a debate, someone says 'but God doesn't even exist' without anyone specifically claiming that He does. In this scenario it is not up to everyone else to prove the guy wrong; he's the only one who actually made a claim about God, so he's the only one who has to prove anything. As far as the other guys are concerned they were just expressing opinions rather than intended truths. If you can't prove 'God doesn't exist' then don't claim it in a debate scenario. Simple as. If you have to say something then state your opinion because you don't need to prove your opinion.

Nobody should claim that God doesn't exist. That would be a stupid assumption. God cannot be disproven so don't claim that try to claim that he doesn't exist. What you should do is look for evidence of his existence. If no evidence exists then you should dismiss the claim that a God exists because it would be irrelevant to talk about it without evidence.