Yes, any driving school vehicle (unless it meets the exemption conditions) needs a valid e-vignette if used on Czech toll roads.
Motorbikes are generally not subject to an e-vignette.
Vehicles with a maximum permissible weight of over 3.5 tonnes (incl. combo of vehicle + trailer) use a distance-based toll system (MYTO CZ) instead of the e-vignette.
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