Not all tig welders use high frequency(HF), some use it to start the weld, others use lift start, some need it to keep the beam going and such when AC tig welding aluminum. There are purely DC Tig welders that use zero HF. Perhaps some tig welders that do AC can be ran purely in DC and use lift start to start the arc rather than HF. Aside from that, where you put the ground clamp can have a pretty massive impact on your question. The summary of this notion is so long as the controller isn't between the ground and arc it likely isn't going to see any power. No matter what, I like to keep my ground clamp nice and clean(contact points, wire brush) and clamp reasonably close to where I am welding. Some people have noted they felt some current flow through them in situations where they were between the ground clamp and the arc. If you were welding close to where the controller sits on the frame, I'd take it off not just because I'd be afraid power might actually flow through the controller, but because I'd not want to expose it to any excess heat (close is a relative term). Perhaps if you are using the controller as a prop in making sure things fit correctly when you weld you might be able to remove the controllers guts and just keep the casing in place.
I can't really make statements on knowing how HF will impact a controller or other devices, my imagination suggests that would be harmless to anything not powered on at the moment, but I am sure there are tons of things I don't understand.