Usually, when I keep a Unity app running for a long time, it eventually crashes. I am wondering if this is due to Time.frameCount exceeding maximum integer size?
No, that's almost impossible :D Even when your app runs with a framerate of 1000fps it would take 24 days to reach the maximum.
int max = +2147483647 (0x7FFFFFFF)
sec = 2147483647 frames / (1000 frames/sec) = 2147483 sec
hours = 2147483 / 3600 = 596h
days = 596 / 24 = 24d