(Updated with clarifications)
Could a tesla coil be safely built, installed and operated on an active road vehicle which is driving a regular city street? Let's say we want the discharges to be 20-30cm long in the longest direction. Could that tesla coil be actively operating for long periods of time on that car while it is moving, or would this inevitably damage some parts of the car or immediate surroundings (like other cars or electric equipment around that) around it? If it could be switched on/off, then perhaps it would be enough to only make sure it doesn't damage the car itself, while turning it off when too close to other cars or objects.
If there is a safe way to operate it, does that way entail building some kind of electrical cage around the tesla coil? Or would that not help? Or is not even required? Would it need to be a bipolar TC since those don't require ground connection?
How would you even approach this question, what would be hypothetical damage to the car if it was not in fact safe? For the types of the damage that it could cause, is there a way to diagnose, monitor and prevent it? For example if the problem is RF frequency interference, is there a way to shield the electronics from it, or monitor it with some sensor to make sure it is within a certain range?
Several people have asked what the "intend application/usage case is". The usage is to use the tesla coil as a show, for coolness factor. (But are they really ever used for anything else at all? Maybe scientific experiments?) The regular operating environment would be normal city streets.
Best Answer
Unlike the commentators, I don't think there's any inherent reason why this wouldn't work, provided you don't overdo it.
The obvious risks are:
I'm assuming the coil is outside the vehicle (on the roof? In a truck bed?) and of small to moderate size and fixed appropriately. The coil's ground should be properly connected to the bodywork of the car or at the very least to the car's electrical ground.
Strong electric fields move to the outside of metal objects. So fields relating to the coil will be kept outside the bodywork of the car. Petrol cars are already very electrically noisy devices with multiple-kilovolt sparks of their own, so provided you don't inject anything onto the positive side of the car's electrical system the existing shielding should be adequate.
Having a large enough coil to arc to other people's vehicles should be avoided - not necessarily because of electrical risks but because this will be extremely distracting. You may also run into police restrictions on "lights" on the car. And watch out for bridges, overhead wires, and other street furniture.
Oh, and make very sure the thing is off and discharged through a bleed resistor before you go anywhere near a petrol station. I think that might actually be the biggest risk.
A grounded mesh cage for the coil to arc to would be a great improvement in safety.