Let's talk about certain specific abilities and their interaction with Doom.
Doom used to be removed by kraken shell and dark pact, but that has changed now. Kraken Shell and Dark pact will not remove Doom.
Skeleton King's reincarnation as well as Aegis will not be negated by Doom.
You can Doom through BKB, or any type of Magic immunity for that matter, however you don't take the dmg from Doom for the duration of the BKB. If you apply Omniknight's repel after Doom, it negates the damage taken for the duration of the repel.
Doom does not disable Abaddon's ultimate borrowed time.
If Kunkka's Tidebringer is ready to splash, then Kunkka gets doomed, he still has that one Tidebringer. After he uses it, Tiderbringer will cooldown, but be silenced until Doom wears off.
Axe still spins with counter helix while doomed.
Templar Assassin's psyblades still work while doomed.
Invoker still retains his gains from which ever orb he has out. ie wex for ms.
Pudge and Silencer can still gain str and int respectively while doomed.
Any doomed target will not take damage when banished from either Disruption or Astral Imprisonment.
(These are the things I can remember off the top of my head. I might have missed some. Doom disables almost every other passive.)