So, during Perseverence, tell Cullen to start taking the lyrium. There are a few tones you can take when you do this, so you probably want to avoid the mean/cold ones; since I'm romancing him, I took the heart option: this leads the Inquisitor to say that she cares what he wants, not just what's best for the Inquisition, and you get a big decision wheel with the options:
A) "Don't risk it. Take lyrium." (Cullen takes lyrium)
"Don't give up." (Cullen will not take lyrium)
For this test, I took A: the Inquistor says "I can't watch you die. Please, Cullen. Take the lyrium." He acquiesces with a resigned look.
He'll ask for some time afterwards, but the next cutscene is in the armory as he's looking at swords. He seems resigned to his course: not happy about it, but not angry. He will eventually say, "Are you leashed until the day you die, or lyrium takes your mind away?" At this point, you get a big decision wheel with three options:
A) "You chose this life" (Cullen will keep taking lyrium)
"We'll find a safer way. After." (Cullen will keep taking lyrium for now)
C) "I was wrong about the lyrium." (Cullen will stop taking lyrium)
- Choosing A will make him take it forever, and will break off any romance.
- Choosing B results in the Inquisitor saying that there must be a safer way, and when this war is over, "We'll find it. I promise." As far as I can tell from youtube videos, the Inquisitor will use the pronoun "we" regardless of his/her class & specialization, implying that the Inquisitor isn't saying "We'll get off of this together", but rather "I'll stand by you and help you get off of it yourself." Romance can continue.
- Choosing C results in the Inquistor reversing her stance: she says "Then stop!" Cullen is taken aback and says "No, I need to focus on the Inquisition." She retorts that he will, "but on your terms, no one else's." He agrees. Romance can continue.
If you later bring up the fact that you're a templar with him, he will still point out that you may lose your memories. I haven't yet seen any suggestion aside from Cassandra's quote that the Inquisitor might get off of it someday.
If you make him stop taking it completely (I'm unsure what happens if you take the intermediate "take it for now"), there is a mention in the epilogue that other templars followed his lead and got off of it. Since I've never played the game as a templar before, I don't know if the epilogue ever references the Inquisitor being a templar or what he/she might have done about the addiction.