Depends on the PSU.
It can work totally fine for a year or two
It can give you some weird issues (friend of mine had a problem with all USB devices for example with his old PSU)
It could just not power up one day
Or could not deliver enough power
It can have a bad voltage regulation causing your system to be very unstable.
It can produce a lot of sudden shut downs
It can explode destroying the unit itself
It can fry and damage other components
Worst case it can burn your house down