Wall o' text incoming. My bad. It kinda ran away from me.
Dorian is a snob, it becomes super obvious if you bring him along with Blackwall and/or Sera. On the upside the majority of his snubbery seems to be relatively surface snobbery, he's consciously aware of it which is good since he won't let it influence his reasoning overmuch. BUT. There are flashes of more ingrained snobbery.
Firstly when you ask him about slavery in Tevinter he says he hasn't given it much thought and 'it's just the way things were' to which I say: horseshit, Dorian. He's keenly aware of the political and cultural milieu of Tevinter, he has an interest in changing it and I doubt slavery is so completely without dissenting voices that a politically sensitive young firebrand would not hear about it. Pick up on it. Be sensitive to it. Unless he wasn't interested in hearing. Mind you this is not to say he doesn't care about it at all/endorses it wholeheartedly but it doesn't rank as high as his other concerns about his homeland like the shortsighted hording (and abuse) of power. His primary concerns are inherently upper class, understandable but also snobbish. [I'm not even going to dignify the Dorian-only-cares-about-changing-Tevinter-because-he's-gay thing because it's just really, really dumb]
Second would be some of the stuff he says. For instance 'elves running Haramshiral' and some of his banters with Blackwall specifically. These things aren't hostile at all, but it reflects a sort of shallow thinking. Surface thinking. Elves running Haramshiral is used as an example of something ridiculous, laughable. For obvious reasons, on the surface, it just seems so damn unlikely to happen. But is there anything inherently BAD about it? WHY is it laughable? Ask him why the abuse or even wide-scale introduction of blood magic could have terrible ramifications and he could probably give you a dissertation ask him why it's laughable that elves should run Halamshiral and he's answer you but he'd have to think some on it more. With Blackwall you can tell he expects the peasantry to act a certain way but doesn't bother asking himself why they do, he just accepts/assumes that they will.
Third, Dorian sometimes comes across as condescending with his interactions with the Inquisitor. Which makes sense if you consider that 3 out of 4 times the Inquisitor has no kind of formal education. I'm thinking specifically of the time he says to Mother Giselle 'you might have to spell it out for him my dear' but there's a lot of other times too. And it's a baseless thing to assume because the Inquisitor is probably highly intelligent (though this is just suppostion on my part based of the fact that the Inquisitor, with no formal education, can educate themselves in Theodosian culture, history, religion and relatively obscure magical theory in less than a year... if you take those perks).
Anyway my point is that Dorian simply does not give the same amount of thought to the nuances and influences at play with the lower classes. So with Briala he probably sees an upjumped peasant, which usually means ambition. Not a wrong assumption but because he sees upjumped peasant he doesn't consider her motives as closely as he should (or the Inquisitor's!), he just sees dangerous ambition. He misses the part where they actually share quite a bit of similarities.
So yeah. Snobbish. Not with intent or a core belief that some people are just better, he's just invested his interest in the floaty upper crust. There's not much consideration left for the lower parts. If you HAVE to be a snob I'd rather you be this kind than the 'some people are just born superior' kind.