It happens very quickly, so you can miss it if you have your attention taken away by the timed dialog response that pops up in your face. Keep in mind that you can pause (at least on PC) when these options come up, giving you time to think of a response, and in this case scope out the scene.
I'm not sure if you have to actually enter the room for the scene to trigger. That said, as soon as you open the door look ahead near the fireplace. Connor immolates himself about a second after you enter. He does this on purpose to prevent himself from being taken over by the demon. Dorian's comments that Connor "held out as long as he could" and that it was "an act of courage" confirm this also. Of course, that's just Dorian's perspective, but I tend to agree.
Also, be sure that you actually talk to Connor in Redcliffe before you go on this mission. You can find him by the docks talking to another mage. The first time I played I didn't know that grown Connor was in the game and on getting to this part in the mission was rather confused. I knew who Connor was, but it made no sense that my character would.