Plasma damage to body cells
Although the voltage of Tesla coil is comparable to that of hard X-ray machines, the construction is not, they are not built in a vacuum. At atmospheric pressure, electrons do not have the mean free path available to get to high enough energies to emit significant amounts of X-rays when they hit the electrodes. Their energy is constantly being dissipated at lower levels keeping the arc plasma hot.
There are dangers from TC coils though.
Breathing in the ozone and nitrogen oxides generated by the sparks is not good for your lungs, so don't run one for a long time in a closed room.
Several operators of Tesla coils have died when sparks, that is plasma channels, have contacted them at the same time as contacting conductors at mains potential. Whereas the current in the sparks themselves is very low, once they complete a route from a low impedance sources of energy, a large current can flow, and kill you if you're part of that route.
At the frequencies involved, humans tend not to feel electric shocks resulting from spark current through their skin. It can still cause damage to surface and deeper layers of body tissue by heating, so-called 'RF burns'. Nerve endings are likely to be the first things to stop working.
UV light from the arcs is not a good thing to watch for a long time.