1) You've done a playthrough of any bioware game where you just don't romance anyone at all
Yes- pretty much every Bioware game I've played I have done at least one playthrough, if not more, with no romancing. (Caveat: still working through SWTOR's class stories.)
2) Is it fun? Would it be fun?
It can be. It depends on what you bring to the game, and what you want out of it.
My 'canonical' maleShep, for example, never romanced anyone because, well, a military leader romancing a subordinate is a bad thing on a number of levels- bad leadership, bad team dynamics, bad professionalism, and more. There was no romance, and you know what? The story worked as well as it did without it.
I think ME was a bit weak in the department of not having no-romance equivalents to romance-dependent emotional support scenes like, say, the Normandy lockdown scene, but DA has avoided those scenes. In DA2, for example, Aveline will come give you some friendly comforting after the mother's death.
Given how little the stories actually ever acknowledge romances, ignoring the lack of them is pretty easy. For characters that don't show an interest unless you initiate, the topic never comes up. For characters that initiate by showing interest, handling that one-sided crush can be a valuable role-play opportunity.
3) How would it play in Inquisition? How would the story play out if your inquisitor never finds love?
As a daring and dangerous saving the world scenario, I would imagine. The same as an Inquisitor who does find love. You don't need nookie to save the day.
You'll have big problems to solve. You'll make friends out of people you don't bang or get banged by. You'll have touching and heartwarming character arcs that make you enjoy the character (or not), make Big Decisions influenced off of what NPCs do or do not approve of, and enjoy some characters over others. Companions will have their various dysfunctions that you may or may not help them get over.