I’ve been riding my fixed on the trainer on and off and it’s been OK, with one proviso. Gearing is important - if the ratio is high then you may not be able to ride at a low enough speed to keep the power low in the valleys. My discussion thread is here:
Right now I’m using the geared bike on the trainer because the fixed trainer gearing needs a different length chain and I can’t be bothered to swap wheel and chain every time I want to ride a TT!