-spoiler warning-
In my opinion, the game implies a deep bond between the two at the beginning. There are some hints at a bit of some sparking romance at the Ordon spring scenes. I think there was still something there after the game was beaten as well.
I'm not denying that Link may have developed some feelings for Midna, but I think he still liked Ilia at the end. For the record, I like LinkxMidna, but I like LinkxIlia better for many reasons that I won't bore you with here. XD
There are many little hints pointing to Link's feelings for Ilia throughout the game. (The way he looks when the mayor asks for her whereabouts, Midna's snarky comments when following her scent, etc.) As for Ilia, she shows her feelings when she asks Link to come back safely.
From the memory restoration scene, you can tell that they both really care for each other (read: looooove XD) and so I don't think Link's feelings faded at all. Ilia's emotions in this scene further show how much regaining the memory of Link means to her. She even sheds a tear! :')
Plus, in the ending credits, you see Ilia watching as Link rides away into the woods. What he's going off to do seems open to interpretation, but the way I see it, she confronted him before leaving and told him to "be careful" once again. She has her hand over her heart, further implying romance. (And according to Hyrule Historia, Link DOES return home to Ordon, so he eventually returns to Ilia in the end! ^^ )
Sorry for describing some of the obvious things in the cutscenes, but these moments are why I think their feelings stayed throughout the game.