Hard to give a straight, unqualified answer here.
If you make a claim which you cannot back with evidence, there's no fallacy in your interlocutor's pointing out the fact.
If you make a claim which you can back with evidence (by standards pertinent to the subject) but which your interlocutor will not allow as evidence, your interlocutor is not perpetrating any kind of fallacy but may be cognitively incompetent.
Here's an example - unfortunately using the out-dated term, 'lunatic' :
A certain lunatic is convinced that all dons want to murder him. His
friends introduce him to all the mildest and most respectable dons that
they can find, and after each of them has retired, they say, 'You see,
he doesn't really want to murder you; he spoke to you in a most cordial
manner; surely you are convinced now?' But the lunatic replies 'Yes,
but that was only his diabolical cunning; he's really plotting against
me the whole time, like the rest of them; I know it I tell you'.
However many kindly dons are produced, the reaction is still the
same. (Henry E. Allison, 'Faith and Falsifiability', The Review of Metaphysics, Vol. 22, No. 3 (Mar., 1969), pp. 499-522 : 502.
No matter what, it will always be the case that 'you have no evidence' against
the person's conviction - claim - that all dons want to murder him. But that is his problem, not yours.
If you make a claim which you can back with evidence (by standards pertinent to the subject) but which your interlocutor will not allow as evidence, your interlocutor may be perpetrating the fallacy of self-sealing argument. That's to say, his or her own argument or argumentative position may be vacuous in the sense that no evidence can defeat it : it is self-insulated from critique. It's the argument that's wrong here and not as in the previous case the arguer.
This might serve as an example (I'm not great on dialogue):
'I have a special fairy who always makes the best things happen to me.'
'But last year you broke your leg and this week your house burnt down !'
'Ah, but that must have been what was best for me, to save me from a worse outcomes.'
'How do you know ?'
'I have a special fairy who always makes the best things happen to me.'
Now, 'But last year you broke your leg and this week your house burnt down !', looks like pretty good evidence (not conclusive of course) that the special fairy does not exist. But it cannot break in on this self-sealed argument. Nothing can. Whatever evidence of sub-optimal outcomes you produce will always be deflected or deflectable by the 'explanation' that those outcomes were not sub-optimal because they happened in order to prevent something worse from happening. 'You have no evidence' that can break the seal.