How often are you doing water changes or cleaning the substrate? How big is your tank.
Bettas do not need a lot of air, they breath from the surface. If your tank is cycled, shutting if off will kill off some of the good-guy bacteria. If it is not cycled, shutting it down won't hurt anything. But, with no filtration, you will have to do 100% water changes instead of partial ones done in a cycled tank.
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First, Do No Harm!
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