All of the high voltage equipment I have worked on was well-shielded, and often insulated with a dielectric gas or liquid, so the electrical field was confined and did not have any effect whatsoever on nearby equipment. This is true in the static, steady state. If arc discharges are involved, which is what you have with a Tesla coil, all bets are off. Not only is it virtually impossible to shield the high voltage arcs (because the whole point is create spectacular arc discharges) but the rapid rise times of the currents involved create considerable radio-frequency emanations. These can be difficult to shield sensitive equipment from, although it can be done. It all depends on what is a tolerable field strength for the equipment involved.