Depends on the type of media and the type of damage.
For optical media, if the damage is due to dust or dirt on the disc surface during burning, the damage will probably not increase over time. If the damage is due to scuff marks or scratches on the disc surface, my guess is that it might not increase over time - except if the circumstances that created the scuff marks or scratches continue to exist.
For optical media, if the damage is due to a bad disc, the damage is likely to increase over time. If the damage is due to a bad burning drive or bad compatibility, the damage may increase over time and then again it might not. This seems to vary.
For a harddrive, damaged sectors will almost always increase - sometimes very rapidly, and you should back up any data as fast as possible and stop using the damaged harddrive.
As a matter of fact, what is the difference between a damage and a bad sector?
From my limited experience you can have damage sectors that won't give you any perform problems at all. But that doesn't seem to be the case with bad sectors.
In a Nero CD-DVD Speed / DiscSpeed ScanDisc Read test, a bad sector is an area that could not be fully read during the test while a damaged sector is an area that caused the drive to re-read data because it wasn't fully readable the first time.
In a Nero CD-DVD Speed / DiscSpeed ScanDisc C1/C2 - PI/PO test (which I don't recommend), bad and damaged sectors depend on the amount of parity errors/failures encountered during the scan. I consider this test less useful than the Disc Quality Scan and the ScanDisc Read Test.
What about the "verify written data", is it useless? Many problems Disc Scan detects the "verify written data" simply ignored. On the other hand, real crappy discs were pinpointed to me be "verify written data" in first place...
Depends on how the verify is done, which is different between burning programs. Many people seem to have little faith in Nero's verify option (which has never steered me wrong).
The verify option in ImgBurn is in my experience trustworthy, especially since it will by default eject and reload the disc after burning. I use verify for ALL my burns with ImgBurn.