We've already seen this play out to a certain degree with the proof of the four color theorem[1]. Basically, the problem was reduced to checking about 2000 specific graphs (the checked property was more complex than simple colorability). The reduction part was very complex but still checked by humans, but checking the ~2000 graphs was a computationally expensive process that took computers thousands of hours to complete, which was completely infeasible for direct human verification. There was a lot of controversy over whether this counted as a proof. Since then the proof has been simplified to 600 special cases and run through a proper proof assistant (Coq), so it is now pretty widely accepted as proved.
It's not quite analogous since the core of the proof was still human created and checked, and the basic form of the computer generated portions was well understood (i.e. the kinds of steps they used). There is some extra doubt involved when this is not the case, since it's possible the computer came up with a novel proof of inconsistency if we don't know at all how the individual parts of the proof work.
Even if someone formulated an apparently human-independent mathematical statement of "interesting", you would still the human judgment that this was credible.
How would this be so? The sheer length of the proof rendering it unverifiable in a human lifespan?
Shinichi Mochizuki spent several years compiling a large body of work which he calls "Inter-Universal Teichmuller Theory" which which implies a number of important results in number theory, algebraic geometry, and other areas (if I understand the gist of it). However, he has run into a wall in the mathematical community in that almost no one wants to spend the time to try to understand what he wrote (I believe he said he believes it would take someone well-versed in the field around 6 months of study to get a grasp of it).
If a human could produce enough work in sufficiently dense terms that other mathematicians don't want to touch it, I can imagine a computer could generate exponentially more work in a much less human-readable format than Mochizuki (albeit formally correct).
I would venture to say "no". I don't think humans will ever have no place in mathematics, because the problems we deem "important" are often relatively arbitrary. If a "post-human mathematician" starts spewing out thousands of pages of mathematics a second, all in a form only a computer can understand, no one will care. If a computer fells a tree in the woods, it doesn't make a sound.
I wouldn't discount the possibility, though, that a future "creative" computer manages to produce a proof indecipherable to humans of a theorem we care about, at which point I think there will be quite a perturbation in the mathematical community. If a computer proves the Riemann Hypothesis in such a way that no one can understand it, but it spits out a Coq document that everyone can load and verify, will people consider the problem solved?