The longer we go without finding some particular piece of evidence that is supposed to provide proof for a hypothesis is, in my view, reason to downgrade confidence in that hypothesis only insofar as we have reason to think we should find that evidence over time.
For example, some scholars hypothesize that the Oracle of Delphi of Ancient Greece ingested psychoactive substances; however, we don't have any concrete evidence one way or the other. Our confidence either way need not change even if we pass a thousand more years without any evidence, as it is reasonable to expect that we might not find any archeological evidence that would bear on the question.
Essentially, I think, our confidence in some hypothesis should only be affected by time insofar as something is happening over time that bears on the evidence for that hypothesis. If someone hypothesizes that Atlantis exists on the ocean floor somewhere and over time we're exploring more and more of the ocean floor without finding Atlantis, that would be reason to downgrade our confidence of the existence of Atlantis.
These are only similar to the question of God's existence if God's existence is dependent on the kind of empirical evidence that empirical evidence bears on. There are problems in mathematics that some mathematicians are confident have solutions, just ones we don't know yet (e.g. the Riemann hypothesis).
Or, think of prime numbers, where for the highest prime number we currently know mathematicians are confident there is another higher prime number even if we don't know it yet (evidence). No matter how much time goes on our confidence can remain unchanged.
I think the answer to your question turns on whether evidence for God's existence consists of empirical facts whose discovery is to be expected over time (e.g. arguments from design which held that nothing could explain the complexity of biology except God which were supplanted by evolution) or if evidence for God's existence is more akin to mathematical conjecture, where some believe we have good reason to believe that God exists on the basis of various arguments even if we are waiting for an exact proof. If we are looking for evidence of the latter kind, I don't think the passage of time would necessitate a downgrading of confidence.
P.S. I should say, though, that many philosophers do think we already have evidence of God's existence, empirical and conceptual (see the various sides for and against here, here, here, here, or here).