With that idea I could say Atheists make the claim "God is not real" and Christians are saying we believe in God, there is no claim there. So the Atheist has to give evidence not the Christian. It depends on whether you want to argue or learn.
Atheism does not state "God is not real".
Weak atheism is the position wherein a person
doesn't state either "God is real" nor "God is not real".
Strong atheism is the position wherein a person states "God is not real".
So you have:
Theism: I affirm the statement "God is real", and reject the statement "God is not real"
Strong atheism: I affirm the statement "God is not real", and reject the statement "God is real"
Weak atheism: I reject the statement "God is real", and reject the statement "God is not real".
Thus, atheism
in general does not state "God is not real". Of all atheists, only a small minority are
strong atheists - the vast majority of atheists are
weak atheists.
So, no, atheists
don't make the claim "God is not real" - if they did, they'd be the minority of atheists called
strong atheists, and the onus of proof would indeed be upon them.
But
weak atheists, which make up nearly the entire ranks of atheism, do not make any claim regarding the existence or non-existence of God. They merely, academically, shrug their shoulders and say "We don't know".