Steamvaults

I believe the next instance up in the queue is the Steamvault branch of Coilfang Reservoir.  These are the associated quests:

•  [70] The Warlord's Hideout Given by Watcher Jhang at the entrance to Coilfang Reservoir.
•  [70] Orders from Lady Vashj Starts with a random drop of the same name inside the instance.
•  [70] Underworld Loam Given by David Wayne within Terokkar Forest.
•  [70] The Second and Third Fragments Given by Khadgar at the Terrace of Light in Shattrath City, this is step 6 in a quest chain to obtain the Master's Key that starts with Archmage Alturus at the entrance to Karazhan in Deadwind Pass. The Second Key Fragment is found in this instance. It can be located in the water just in front of the first boss Hydromancer Thespia.

So it looks like Underworld Loam and The Second and Third Fragments are the only quests that can't be picked up in or near the entrance to the instance.