I take a more pragmatic position. For me, the burden of proof lies with whomever wants to change a mind.
Yes, I know what you're saying here. It's only reasonable to ask the person making the existential claim to furnish evidence and argument.That doesn't work. Except in those few cases where the description of an entity leads to a logical contradiction, it's not possible to demonstrate that an entity does not exist.
For example, there is no evidence that you could produce to demonstrate that there is *not* a china teapot in orbit around the Sun between the Earth and Mars. Do you seriously think that those who claim that such a teapot exists and those who reject the existence of such a teapot have an equal burden of proof?
Therefore, it is reasonable that the burden of proof would fall upon the person claiming that some particular entity exists, and not on "whomever wants to change a mind".
I think that you're being a little too concrete here and defining burden too narrowly. It appears that nobody feels any burden about teapots in space one way or the other.
But if you did - if you felt the need to change minds of others about the teapot - there's your burden, and you'd better get busy trying to convince them, whichever side of the issue you choose. For example, if a trillion dollars has been allocated to finding the teapot, those who disagree have the burden to act. Or just sit back and watch.
It is in that sense that I say that the church has the burden of proof - not necessarily to prove that their god exists, but that to convince its detractors that their church should be supported and preserved.
It's only a coincidence that the theists happen to be making the existential claim in this matter. If the situation were reversed, and irreligiosity were in the process of being supplanted by fundamentalism, it would be our burden to act, not theirs.
That's what I mean by pragmatic. The one who wants to change minds has the burden to convince them. When somebody asks me to prove that their god does not exist, my answer isn't the philosophical one that it is impossible. It's the pragmatic one, "Why bother? Your religion is dying out."