Good question, from my own experience in dota, I'll just throw a list that might help you.
Here you go:
Silencer: Imagine playing Invoker or Storm spirit against him. You'll get silenced for every skill you use, which happens A LOT for these heroes
Hopefully, a simple counter for this is a magic stick to get some mana/hp whenever he casts his DoT (for early game) and BKB (black king bar)for his ultimate/passive aura (mid-late game).
Doom: This hero can make ANY hero almost useless by just casting his ultimate on him.
His ultimate will just shut your skills(and items) for a huge amount of time.
Even some passive skills won't be triggered when you're doomed (like Phantom Assassin's ultimate). A counter to him is getting Linkin's Sphare or disabling him and killing him before he uses anything.
Pugna: Actually he's not as destructive as Doom or Silencer, but he can shut down any caster with his zap wards. A support Lion(in competitive games) who has about 600~800 HP will instantly die by using his own skills, this applies to other heroes who consumes lots of mana, like Invoker and Lina.
Destroying his ward will solve this issue - otherwise a BKB is a good alternative.
Invoker: This hero has a whole set of skills which means there's no specific counter for him - this fact is enough to make him a counter hero. Besides he has lots of CC (disables) skills, which are somehow unique and very effective. Invoker is tricky to counter, because BKB doesn't help much since Ice Wall will slow you down even though you're BKB'ed, this also applies to the blast (I'm not sure though).
other heroes with disables: Almost all heroes with long disables are considered counters, heroes like Earthshaker (with aftershock), lion, and rhasta are counters to many heroes, especially fragile heroes who can't survive for enough amount of time to do their DPS.
The old school counter for these heroes is getting a BKB, otherwise stay behind till they consume their skills, or just kill them before they do anything.
I'm open for your opinions.