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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.
Bleed effects
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.
Audio cues
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.
Heal effects
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.
Other
Though a disabled vehicle's movement is impaired, its weapons function normally.