The burden of proof is on whomever is making the statement. I can say green swans exist. You cannot prove they don't. Therefore you can only say you don't know. By making the assertion that they don't, the burden is on you to prove it. Green swans exist is an verifiable, unfalsifiable statement. The opposite is unverifiable and falsifiable. Making a definitive statement which is unverifiable is silly, don't you think?

The burden is on the atheist and the theist both. It is not on the agnostic.