I am solely speaking through my own experience from my playthroughs.
I mostly play as male Sheppard.
In ME 1 I had a romance with with Ashley Williams, going all in on her.
In ME 1, as male Sheppard, you are faced with a choice between Ashley and Liara, where I chose Ashley. Liara will still have a crush on you which you can see through her dialog.
My romance where Ashley Williams.
(Kaiden was sacrificed no wonder)
In ME 2:
you join Cerberus. Ashley Williams is Alliance to the bone, and wont have anything to do with you, despite that she misses you when you encounter her on Horizon. So in ME 2 I had my romance with Miranda Lawson, going all in.You also encounter Liara on Illium who deep within still have a crush on you. But you are nothing more than friends here.
In ME 3:
Sheppard joins the alliance again. And Ashley Williams rejoins your crew. But, she heard about the romance from ME 2 with Miranda Lawson. She is by no doubt "pissed" about it as she still loves Sheppard. BUT, when Ashley Williams gets into the hospital, you have the option to reengage in a romance.I cant remember the exact dialog which gives you the possibility or reengaging in a romance with her.
But to conclude on your question.
YES - it is possible to continue a relationship from ME 1 in ME 3.
I can't give you detailed information regarding other romances, only that it is possible with the named characters.