My best friend in middle school through high school fell for me, the feelings were not mutual. He was there for me through a lot of stuff and eventually threw his support in my face as a reason why I should "see that he's a great guy" and "give him a chance". I did not, and it hurt me a lot that what I thought he'd done out of friendship he only seemed to do because he wanted to date me or use it against me. He also spent a lot of time nitpicking any SO I had and basically saying they weren't good enough for me. Whenever I broke up with someone he was there immediately which was nice at the time to have that support, but looking back it seems more like he was hoping I would run to him when things didn't work out with someone else. We haven't talked in years, after he blew up at me and got mad at me for not dating him I tried to talk it out with him but there was no salvaging it.
Two of my other friends had feelings for me. One was in high school and the feelings were mutual. We didn't exactly date because she was afraid to come out and wanted everything to be on the DL so we basically acted like best friends but kissed a bunch and fooled around in private. Then she moved away.
The other was a friend I'd made in college, and the feelings were not mutual. She was understanding that I was in a long-term relationship already and didn't have those feelings for her. I stayed friends with her for a while but she got drunk at my birthday party and tried to kiss me. She never apologized for that and we slowly stopped talking.
Do I think the friendship can be continued? Yes, I've had unrequited feelings for friends before and though I never outright told them, I would gauge their interest and if it wasn't there I would back off and just stay their friend. I think it depends on the person, though. If you feel that you can't continue the friendship without making things weird for the person you have feelings for then you should probably step aside until the feelings pass. You may not be able to control how you feel about someone but you can control how you act towards them and can show them (and their current SO, if they have one) respect by not making things difficult. Also half the time it's infatuation anyways. Actually dating someone is different from being their best friend and you'll see a new side of them that may not be the fantasy date you built up in your mind. But yeah, I think it can be continued, just tread lightly and be respectful of the fact that they don't feel it too.