Do I have to pay road tax if my car is a classic car?

Road Tax exemptions for cars (even for classic cars) only apply if they were built prior to 1st January 1981 on a rolling basis.
To be specific, cars are exempt from tax as of April 2021 if they were registered prior to 8th January 1981, so it's really 40 years and 3 months give or take a few days.

Car tax exemption does not apply if you are hiring out your vehicle, classic cars used for weddings for example, or other commercial pursuits.

Take note, you do need to apply to stop paying your car tax and the DVLA will send you a vehicle tax reminder letter before your tax is due to expire. You’ll need to tax your vehicle, but will not need to pay and it's an initial one-off payment of nothing and that's it, you won't have to worry about renewals theirafter.  

When can you stop getting an MOT for your car?

You do not need to get an MOT if your vehicle was built or first registered more than 40 years ago and no ‘substantial changes’ have been made to it in the last 30 years, for example replacing the chassis, body, axles, or engine to change the way the vehicle works and you don't have to apply to stop getting an MOT for your vehicle each year, but regardless you still have to keep it in a roadworthy condition.