Can an AC and a dryer go on the same breaker?

Without understanding the current rating of the wiring, then this question can not be answered safely. The purpose of the breaker is to limit the current well below the current rating of the wire, so as to prevent overheating (fire) risk.

If the wire is rated to 31 Amperes and a 60A breaker is installed, then it is conceivable (if not inevitable) that a load exceeding 31A will cause overheating (fire).

At the risk of being unkind, I suggest that whatever you do, you consult a qualified professional and not rely on internet advice regarding electrical safety

First of all, you cannot just change out a 30A breaker for a 60A one. The 30A breaker is there to protect the wiring and that is almost certainly wire rated for a 30A max load. Changing to a 60A breaker is a recipe for disaster (fire hazard) and highly illegal.

Additionally, high current loads like a dryer or an HVAC unit must be wired on their own circuit. So your plan fails this requirement also.

Rather than do this, why don't you just run a new circuit for the AC unit and do it right? That, of course, will involve pulling a permit from your local authority and having it inspected and approved for use.