After a quarry has finished its dig (reached bedrock or lava), will the chunks be unloaded or will they remain loaded until the quarry block is removed?
Yes, a quarry will keep chunks within its area of operation loaded before and after doing its thing. They will only be unloaded when the Quarry block is removed.
To test this, I simply set up a system that pumps items out of an Ender Chest (from the Ender Storage mod) into a regular chest near a quarry that had reached bedrock.
Then, I went to the Nether, restarted the server for good measure, and put items into the corresponding Ender Pouch. They started disappearing, which means that the chunks in the overworld were still loaded.
Which version of buildcraft are you using?
In my version, that's in thekkit 3.1.2 (that's based on minecraft 1.2.5) a quarry does not keep any chunks loaded. You have to place world anchors to keep it working even if you're away (or offline).