Now, in a sense, this is entirely correct. Theists do have a burden of proof. And if all that these atheists are saying is that they lack belief because they have not yet seen any good evidence, then it is in fact their opponents' job to make their case.
However, many atheists who make the above claim also often say that God doesn't exist. But to say that God doesn't exist is also to make a claim, and thus to incur a burden of proof. And yet, when challenged to justify their positive disbelief, some of these atheists revert to the “atheism is merely a lack of belief” line, and deny that they have to justify anything.
The mistake here is obvious – almost too simple to seem worth pointing out – yet is one I've seen several times online. And it just opens the door to criticism of atheists.
Perhaps there is some confusion in all this as a result of the oft-repeated claim that “one cannot prove a negative”. If one cannot prove a negative, and the statement “God does not exist” is a negative, then one cannot prove “God does not exist” – and doesn't it then follow that one cannot have a burden to prove such a thing? I've seen at least one YouTuber apparently arguing along such lines.
But, to begin with, a burden of proof – in spite of its name – is only a burden to provide justification for a claim, and not strictly speaking to prove it. (When someone makes a claim, they should be ready to provide a reason for accepting it as true. As a general rule, however, it is too much to demand that they be able to prove the claim is true.) Furthermore, it simply isn't the case that one cannot prove a negative, so this entire argument rests on a confusion anyway.
My suggestion is that nonbelievers need to make a clear choice between so-called positive and negative atheism. Positive atheists are those who make the claim that God does not exist, whereas negative atheists merely lack belief in God. Positive atheism, in other words, is “traditional” atheism – the belief that there is no God; negative atheism covers all other nontheists.
If one is merely a negative atheist, then one can place the burden of proof on one's opponents. As soon as one makes a claim that there is no God, however, then one has incurred a burden of proof.