The burden of proof is on whoever makes a CLAIM. A claim is an attempt to require someone else either to adopt what one asserts, or to act in accordance with it.
Simply stating a belief is not a claim.
Calling someone stupid or deluded or foolish for holding a belief, however, is an indirect way of imposing a claim.
In each instance, therefore, it is important to review the language used by the other party, to see whether they have effectively made a claim and incurred the burden of proof. That's not always the theist, by any means.