TKCs can indeed affect the steering feel at speed, but I've found it more of a wobble/weave than a shake though (on a 21" wheel). In my experience, a bit of fiddling with tyre pressures can alleviate this, or rather I should say, move the point at which the wobble starts further up or down the speed range.

When you say shake, do you mean a high pitch rapid twitching or vibration felt through the bars? (Rather than a weave.) Is this oscillation violent or gentle? How worn/old are the tyres?

Could be a multitude of other things, such as: knackered or incorrectly adjusted head race bearings, poor damping (front or rear), unbalanced front/rear ride height, damaged rim, wheel(s) out of balance, loose spokes etc...

__________________