Burdens of proof do not exist in philosophy, and insofar as they do, do not track, e.g., the distinction between positive and negative existential claims. The idea of the “burden of proof”, as it is used today, is mostly an artifact of the sloppy folk reasoning that goes on in New Atheism (and of popular but fallacious slogans like “You are never called upon to prove a negative!”, courtesy of Ayn Rand).
All positions carry a “burden of proof,” whether or not they are positive or negative, since all propositions must be justified with reasons and evidence. For instance, atheism is the proposition that “God does not exist.” As a result, many atheists will say that “Because it is a negative claim, I do not bear the burden of proof; instead, the theist does, because the theist is making the positive claim that God exists.” But this is completely wrong (and awfully convenient).
Rather, the “burden of proof” for atheism can be met; there is no need to pass it off like a hot potato. For example, the proposition “God does not exist” (let us label it G) is justified by our best scientific theories. That is, not-G contradicts our best scientific theories—theories for which we possess a lot of evidence. Therefore, we have strong reason to believe that G.
If there is any such thing as the burden of proof in philosophy, it has to do with what’s called a “dialectical burden of proof,” which is nothing more than the following: Reasons and evidence have been provided for x. You hold ~x. Unless you can provide us with reasons and evidence for ~x that outweigh the reasons and evidence we have for x, then we rationally ought to hold x.
Again, insofar as there is such thing as a burden of proof, it is not drawn with a view to negative vs. positive statements, nor anything having to do with either x or ~x being inoculated from justification.
Indeed, if “faith” is supposed to be belief in something without evidence, then those who declare that they do not bear the burden of proof are engaging in a clear exercise of faith, for they are literally saying that they are entitled to believe a certain proposition (e.g., “God does not exist”) without any reasons and evidence required to justify it.