You sort of have to wonder whether poor weather was not uncommon during the great Dionysiac festivals, since several of them occurred during the winter rains. The Rural Dionysia took place in Ποσειδεών (roughly December, the rainiest month in Athens) and the City Dionysia took place during Ἑλαφηβολιών (March/April). The Lenaea took place during Γαμηλιών (the "wedding month," roughly January) and the month actually was known in the other Ionian calendars as Ληναιών. I can think of only one occasion on which bad weather is mentioned, though. Plutarch mentions that when the Athenians voted certain honors on Demetrius, including renaming the Dionysia to the Demetria, the gods responded with divine prodigies of their displeasure. Plutarch specifically mentions that:

τῇ δὲ ἡμέρᾳ ᾗ τὰ τῶν Διονυσίων ἐγίνετο, τὴν πομπὴν κατέλυσαν ἰσχυρῶν πάγων γενομένων παρ᾽ ὥραν.

But on the day on which the Dionysia was, they cancelled the procession with unseasonal ice [I guess he means frost] arising

Plutarch makes no mention of the actual dramatic performances, but the Dionysiac festivals were not only dramatic--at the City Dionysia the theatrical performances only began on the third day, after two processions, one bringing the god's image to the orchestra and the other including sacrifices to the sanctuary of Dionysus (this larger procession on the second day is what Plutarch is talking about). Whether the Dionysia simply proceeded without the procession is unclear, although personally I find it doubtful. But Plutarch seems to think of this as a very unusual situation, and I know of no other instance of any part of the Dionysia being cancelled, so presumably the situation rarely, if ever, arose