I think we use de mâine when it's a recurrent thing that has not yet began or had a break, like school or a TV series. And we only use mâine when it's something happeing tomorrow or just tomorrow, like a concert, or fixing the door.
So we would say:
- Începem școala (de) mâine.
- Începem să reparăm ușa mâine.
- Începe telenovela (de) mâine.
- Începe concertul mâine.
Also, note that in a way, school starts everyday. So before each school day, I can say școala începe mâine but only before the first day of school I can say școala începe de mâine.