Yes, there is no exemption for classic or vintage cars in general. Unless the vehicle meets the conditions for exemptions (e.g. as for disabled people, zero emissions, etc.) an e-vignette is needed to be used on Czech toll roads.
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.
Related articles:
What vehicles need a Czech e-vignette?
What happens if I drive without a Czech e-vignette?
Where can I buy the Czech e-vignette?
What is the MYTO distance based toll?
Where can I register for the MYTO CZ and get the on-board unit (OBU)?