Stating first of all that, of course, the question of how the Tonys and other theater awards will deal with this crisis is far from the most pressing concern, here are my brief thoughts on the subject:
It seems to me that, even under the best case scenario involving COVID-19, the number of shows opening through the fall will very likely be quite low. This fact, combined with the cancellation (or indefinite postponement) of so many shows that were to open this spring, leads me to think that the best course of action will be to reschedule the Tonys for some time in 2021 -- maybe in January, if things (hopefully!) go really well, or maybe not until June 2021, if the recovery of the theater industry takes much longer. And then, in the record books and databases, there will just have to be asterisks next to the names of all the nominees and winners.
If the Tonys were to be rescheduled earlier -- say, fall of 2020 -- that would be problematic in the sense that there would be far fewer potential nominees than usual. Of course, if they're rescheduled later, that's problematic in the sense that there will (hopefully) be more potential nominees than usual. But all things considered, including logistics and many other factors, I would say later is better.