Asserting the burden of proof lies not with the person making the claim, but with someone else to disprove.
This example also invites a moving of goal posts fallacy, as many people making this argument will never be convinced their belief has no basis in evidence, no matter how many times you debunk their so-called “evidence”.
This one’s a biggie. Many people fall for this trap as it is easily hidden within a larger, factual, claim. Make sure every assertion is backed by evidence before allowing a discussion to advance. Letting this one slip will mean you can quickly be backed into a corner, no matter how correct your position.
A popular example of this, useful on those who have trouble getting their heads out of their religious bubble, is by claiming an invisible teapot is, at this very moment, in orbit around the Sun between the Earth and Mars and that because no one can prove the assertion wrong it is therefore a valid one.
This is, of course, nonsense. How would I know there is a teapot there without evidence? It is my burden to provide such evidence.
Atheists can’t prove God doesn’t exist!
Nor do they have to, just as religious believers don’t have to prove Santa Clause or everyone else’s god doesn’t exist, neither do atheists. The onus is always on the one making the positive claim.
Evolution is a lie, made up to deny god! God created man.
Interesting proposition. Is there any evidence to back it up?
Nothing besides a book and a bunch of people who think the book is correct. Evolution deniers will often cherry pick disproved hypotheses and falsified fossils while ignoring the mountains of valid, coherent evidence in the hopes people will abandon their acceptance of evolution.
What you don’t see creationists doing is presenting a case for creationism, as there is no evidence beyond their holy book. The dirty little secret of creationists is even if evolution weren’t observed fact backed up by a theory which successfully explains every shred of peer-reviewed evidence ever discovered, they still wouldn’t satisfy their burden of proof for a deity-driven creation, or even a deity at all.
That’s the power of the burden of proof and why you should never let someone slip an unsupported claim into a debate.