I always said Varric probably wouldn't be an option, so this isn't surprising to me.
What's interesting is the question: who is the race-gated romance of the bunch? We used to speculate that it might be Varric if he was an option, but since he isn't, it leaves my curiosity hanging. I hope it's not Viv.
I also hope it isn't Vivienne who's race gated. Enough people already think she's a total witch, and race gating her against elves or something is hardly going to help that. If she is race gated, I'd hope she's race gated against humans, (because she has problems with the Trevelyan for example, not because she doesn't like humans), even if it would ruin my plans.
I suppose one could build a case for most LIs and why there may be story reasons for their race gating. Cullen, (and to a lesser degree, Cass), could well only go for humans since Cullen's already shown religious bias against dwarves and qunari, (that could also extend to Dalish elves). That would kill me though. For my first playthrough I'm hoping to romance Cullen as an Elfquisitor so that would kill me. But then again, they did say they wouldn't force us to having certain religious beliefs, and that would be the big thing with them I suppose.
Sera might not want to romance a Trevelyan Inquisitor since she hates nobles, but I she have to be especially prejudiced to still dislike a human mage.
Dorian might not want to romance a member of the 'slave races', i.e. elves and qunari. I don't believe we've heard his opinions on slavery, have we? He could be all for it for all we know. I do remember reading that the Tevinter economy would totally collapse if they abolished slavery, so it would make sense.
If he's confirmed, Solas will probably be race gated too. I really love the theory that he wouldn't want to romance a dwarf (or qunari?) because he wouldn't be able to share his Fade adventures with them.
Blackwall is a huge mystery. He could have a very rich past before he became a GW that may stop him from wanting to romance a particular race. Off the top of my head, he could be a former Chevalier and romancing an elf might be out of the question because memories of his initiation ritual would make him feel to guilty?