Yes, driving school vehicles are subject to the same regulations and unless they meet the exemption conditions they need a valid e-vignette. Motorbikes for example are generally exempted.
Vehicles that exceed 3.5 tonnes of permissible weight use the distance-based toll system (MYTO) instead of the e-vignette.
Related articles:
What vehicles need a Slovak e-vignette?
Can I buy multiple Slovak e-vignettes at once for a fleet?
Are commercial invoices provided for purchases of Slovak vignettes?
Which roads require a Slovak e-vignette?
Can I drive into Slovakia without a valid e-vignette if I avoid motorways?