For me this is very simply answered: the question is invalid. To ask it at all is based on fallacious understanding of the issues involved.
I prefer the famous definition of "atheism" often stated as something like "an atheist is someone who believes the evidence of the existence of God is roughly on the same level as the evidence for the existence of werewolves."
The scientific method says you don't prove anything; instead, you disprove the contrary hypothesis (called the "null hypothesis"), by showing that this opposite hypothese is impossible.
For instance, let's say I hypothesize "I think Jim exists." Can we test this theory? Well, to test this scientifically, we'd take the null hypothesis, "Jim does not exist." Can we prove this is impossible? Well, you might call Jim over to tickle me. In this case, I would say, we have significant evidence that the null hypothesis absolutely cannot be true, because, we all just directly observed Jim's presence and disarming affectionate assault; and, what's more, I'm doubled on the floor laughing. So, "Jim exists" is a good theory, and we have met our burden of proof (even though, obviously, there's still room to argue alternative theories. In science, the most likely answer is the best one to go with; until something contradicts it by fitting better with the previously established-and-tested evidence. All you can ever do is prove what correlates best with present understanding, not what's "true". Nothing is ever "true" in science, it's simply "our best available model.")
More realistically, the hypothesis "This new treatment is effective in curing moderate to severe Crohn's disease" is considered valid if we disprove the null hypothesis "This new vaccine is not effective in treating moderate to severe Crohn's disease" by showing a statistical correlation between treatment use and subsequent moderate to severe Crohn's disease remission that is too far outside the bounds of random probability to be dismissed as a chance occurrence. Again, this doesn't provide 100% proof, but, we go with strong enough likelihoods (and in this example, as in many, in the lab there is a rigorous mathematical definition of "strong enough", it's not just a researcher kind of eyeballing it and deciding "looks good enough for me!". Google "confidence intervals" if interested in that tangent.)
However, some statements are not falsifiable... that's to say, it's not possible to come up with a null hypothesis that it is possible to disprove. I might say, "I believe someday a boy will be born who can swim faster than a shark". The null hypothesis "A boy will never be born who can swim faster than a shark" cannot be disproven, because to disprove it, you would have to wait around literally forever to know for sure about every boy ever to be born and check their lap time in the pool. On a practical level it's just not possible to disprove.
So, "I believe someday a boy will be born who can swim faster than a shark" is said to be "non-falsifiable"... it is not a valid question to even discuss "burden of proof" on, at least, not in the scientific sense. It is not a question we can scrutinize empirically, only conjecturally.
A big part of this is measurability by our human sense. For a hypothesis to be falsifiable, the full attributes of null hypothesis must fit within our perceptions, because "disproving" requires disproving every possible case. You must, on a practical level, have a ruler big enough to measure it, and it must be something small enough to measurable within the time constraints of human effort.
So to speak of a "burden of proof" with regards to the infinite, immortal, invisible or immaterial doesn't even really make sense.
Now, I once had a Christian friend give me a really interesting answer to these sorts of questions, in different but related terms. I was needling him about logical inconsistencies in his faith, or some such, and he cut me off, and told me something I've carried with me to this day. He said (something along the lines of) "You're trying to reason me out of my belief. But it's not reason, it's faith. I made a leap of faith. I put my faith in these things being true. So you're not going to reason me out of it, because I didn't use reason or proof to get there. I just have faith in it. "
I had a lot of respect for that answer.
One thing you have to remember, I'm fond of saying, is that reason and faith aren't opposites—they're orthogonal. They're two perpendicular dimensions, having nothing to do with each other. What someone might say is true because it's the best current model that can be arrived empirically at empirically, and what someone has chosen to have faith is true, are two totally unrelated things. We only confuse them with each other because they both get called by the same word: "belief". (It's kind of like how the work of Ravel, Bill Evans, the Angry Samoans and that boring Donavon Frankenreiter song from 20 years ago all get called "music". It's like, yeah, but, really, no.)
So, having started by showing how I think a person of a scientific bent would reject the notion of a "burden of proof" for or against a deity as not even a valid question. We now see that to a person of deep religious faith, the notion of a "burden of proof" is also invalid. Proof has nothing to do with faith.
And to me, as an apizzaist, er, atheist (sorry, I get confused whenever I don't see any evidence of the existence of a deity while I'm also not eating a pizza), there's nothing to prove. The fact that I don't feel I've seen any evidence of the existence of deities isn't falsifiable either. And it's not a belief. Show me the evidence, and I'll change my mind.
I always say, I'm the easiest guy in the world to convince of anything. All you have to do is show me the evidence.
BTW, I used to call myself an agnostic, but a friend made fun of me for it. "Oh, are you going to say now the jury is out on fairies and werewolves? Are you going to say you think they might exist too, just because you can't conclusively prove that they don't?" I had to admin, no, I don't think there's even a chance that any of those things might exist, I'm not agnostic on fairies or werewolves or Odin or Zeus or Pikkiwoki or God or Quetzalcoatl.
But, to refer to a burden on me to prove any of it to someone else, or on anyone else to convince me of anything, is, to me, just a misunderstanding of what both scientific reason and religious faith really are. To try to debate it at all is misguided. It's literally not debatable, from either a scientific nor a religious standpoint.
And on a personal level too, it just doesn't seem like a useful topic to discuss, or even one there's any reason to, whether between two people who agree or disagree, except maybe as a bit of an intellectual exercise in omphaloskepsis. I'm as fine if people want to put their personal faith in things there is no evidence of as they should be (er, as I wish they'd all be) with me not wanting to. I don't see how needing to change anybody else's mind factors into it at all. That's not religion, that's politics. Not that the two have much of a history of keeping their fingers out of each others' drawers. But that's another issue.