There is a logical fallacy that only positive (claims of existence) require burden of proof. But that cannot be further from the truth.
First, burden of proof is more of a system of etiquette than anything else. It helps prevent a discussion from decaying into a back and forth of “yes it is; no it’s not.”
So what issues are there with demanding that only claims for existence require burden of proof? The first is that if this were true, then I could simply claim that no position has burden of proof. That would be a claim of nonexistence. That would put us in a rut very quickly.
But there is another issue. Essentially all claims of existence can be restated (even if in a convoluted way) as claims of nonexistence. Let’s take one of the most argued claims, that relate to the issue of burden of proof: the claim that there is a god. I can restate that as “there is no universe in which a god does not exist.” These claims are logically equivalent, but the latter is a claim of nonexistence. I can also restate the claim that there is no god as an existential claim: “there is a universe in which a god does not exist.” So how can simply changing the way in which we state a claim, change whether it carries with it burden of proof?
The Used Car Issue
Of course, in our mundane life, we would never demand that only claims of existence require burden of proof. For instance, if you were purchasing a used car, and the seller told you that the car had never been in an accident, would you simply believe him, or would you demand evidence? If you would demand evidence, then you are admitting that a claim of nonexistence does, at least in some cases, suffer from burden of proof.