Okay, I'll play. Keep in mind this is what seems to be the most realistic, not what I want personally.
Straight-
Cullen
Cassandra
Varric
Bisexual-
Iron Bull
Scribe Lady 'Scribbles'
Solas
Homosexual-
Sera (confirmed)
Dorian
if I had to venture any restrictions race (or class) wise, I'd have to assume the following:
Cullen- humans and elven women (shows an interest to either a human or elven PC in Origins)
Varric- dwarven women or dwarven/elven women only (says he isn't into humans in DA2 plus with Iron Bull's interview he distrusts qunari)
Cassandra- human men or human/elven men (like Cullen but straight female version)
Iron Bull- qunari only or any non-mage/warrior only (anyone who 'wouldn't break' sounds like he'd only be into qunari or only into strong types (warriors))
Solas- elves only or any non-human
Scribe Girl- no restrictions (if she's the Kelly Chambers of Inquisition)
Dorian- males of any race (doesn't seem like the type who'd have any hangups)
Sera- females of any race (seems very open minded race wise considering she doesn't identify as elven or with human culture)
I don't mind gated restrictions based on gender/race as I can probably drum up a PC that fits the romance arc and overall story if need be.