I want to use a public domain piece found on YouTube but it was recorded in midi form by someone else. I only want sections of it, not the whole thing. Do I need permission?
Yes, you need permission. Only the pattern of notes themselves is in the public domain.
Any arrangement, performance, or recording of the piece is its own creative work - and this recording sounds like it was made somewhat recently, so it would not yet have entered public domain itself. To use it in your game or other work, you would need a licence from the party who owns the rights to that particular recording.
If you recorded your own performance of the piece however, you would not need an additional licence if the source material is in the public domain. (If using electronic samples in your performance, be sure you have the licence to use those in this manner, of course)