There are definitely places where it can be done. The Reimann Hypothesis comes to mind:
The non-trivial zeros of the Riemann zeta function all have real part
This is certainly a claim about existence, and it may be falsifiable in the future. By a loose definition of "ontology," this would qualify.
However, usually "ontology" is not concerned with such things. While such a thing as a zero of the Riemann zeta function could certainly be said to exist, it also exists wholly exists within the realm of mathematics, and mathematics has far more powerful tools to analyzing the existence or non-existence of anything in its domain.
It does seem to me that ontology captures "being" questions which are not accessible through other ways of thought. In the realm of "pure thought," like mathematics, there are usually "better" ways to deal with an issue. It's only when dealing with "the real word" that the perception issues of the division between body and mind become difficult enough that ontology really starts to shine. And, unfortunately, nobody has any reason to believe that problem (which has stood with us for millennia) will be solved any time soon.
There is also the issue of linguistic relativity. If there does not exist a discipline which can solve a particular problem, and then the discipline is discovered, can we say for certainty that the question, as originally posed, meant the same thing to one's predecessors as it now means to you? This may be sufficient ambiguity to create an Ontological claim which becomes falsifiable over time.
One final solution I'd recommend is simply using Ontological terms in situations where Epistemology (and/or science) is sufficient. Consider Schrodinger's Cat. You can make an ontological claim, which becomes falsifiable when you open the door. Or, for a more concrete example, consider a box that has a radioactive atom suspended in it. Put detectors all around it. You can claim "The radioactive atom exists," which is an ontological claim. If you cannot open the box (for some reason) it remains unfalsifiable for now. However, when the atom "chooses" to decay, and that decay is observed, now the ontological statement "The radioactive atom exists" is falsifiable (or at least an interesting enough discussion to start with "falsifiable" and debate the proper adjective).