"It would be a shame not to see you play."
"Not to see you play" is a noun-clause here; it works as a thing. It answers the "what" in the question, "what would be a shame?"
You could also say, "it would be a shame if I didn't see you play". Here, you're using the Second Conditional pattern. This also means it's unlikely you'll miss your friend's playing: you do plan to see him/her play.
"It would be a shame that I won't see you play" is incorrect: you're talking about an imagined situation (would be) and a sure prediction (won't/will not) at the same time. This makes no sense. The best fix for this is to start with, "it will be a shame..." and make the missed appointment definite.