I never got his quest to flag in the game either. After I'd completed the game, I read some stuff online on this (it's a known bug) and went back to an earlier save to see if I could initiate it, and got it to work - the only requirement is that you see a cutscene conversation with him about Felix, after which Mother Giselle will appear almost immediately in the main hall (not the garden) to deliver a message.
The cutscene never happened for me in my first playthrough, even though I checked in with him constantly and had high approval. I thought the cutscene wouldn't be available after a certain point, but another poster (I think in this thread) said they were able to get the conversation and the quest very late in the game.
Posters on the Bioware forum have different suggestions to get the conversation to appear; some suggest not flirting with him or talking too much to him (well, guilty on my part), others say to approach him immediately after getting Skyhold, etc. The bug may also be affected by whether you did the Templar / Mage quest... ugh.
http://forum.bioware.com/topic/530364-is-dorians-side-quest-buggedpossible-spoilers/