I just investigated your question on my server (standard - not hardcore in case it matters) with a couple of M1 Abrams tanks. I haven't confirmed these numbers for other vehicles, so results might be different for other vehicles.
The magic number
The number for disabling looked like 50%. I would have guessed it was that, but I'd never tested to be sure. A 52% tank was fine, and a 50% tank was on fire.
Once the tank was disabled, I had a grace period of a few seconds before health started bleeding. It seemed like about a 10 second period.
But 50% was the point where it started bleeding. Once it starts bleeding you have to either get it repaired or, failing that, bail out before it explodes.
Down to about 60% damage I heard a slow klaxon-like sound (like an
angry little alarm clock).
From 50-60% the klaxon got very fast.
At 50% and below, once the vehicle was on fire, the "woop-woop" alarm sound was added.
When a you repair a vehicle's health to the point that it's no longer disabled, all adverse effects from disabling are repaired too.
When a vehicle is anywhere above a disabled state, after avoiding additional damage for a few seconds it'll slowly regenerate its health back up to 100%. In my tank testing it took about 12 seconds of non-damage before regeneration started.
Though a disabled vehicle's movement is impaired, its weapons function normally.