Does a person claiming God does not exist have the burden of proof?

I agree with the theist. There is a difference between these two scenarios regarding burden of proof:

A:
Person 1: "God exists"
Person 2: "I don't believe that is true"

B:
Person 1: "God does not exist"

In both scenarios, person 1 has the burden of proof because they are the one making the claim. Some try to argue that in scenario B, since the person is making a negative claim that they are free from the burden of proof. This is not the case. The person would be obligated to provide evidence in the form of absence or proof of impossibility.

It is worth mentioning the difference between epistemology and ontology here. Most atheists would say that they don't believe God exists (epistemology) rather than God does not exist (ontology). The former is a reflection of belief and the latter a claim of existence, which certainly requires justification and support.