No that's not right. The FFT is not infinite dimensional. It's the same dimension as whatever length FFT you take. The FFT is just a set of algorithms for computing the DFT. In general they work by recursively decomposing an N-length DFT into shorter length DFT's.

The DTFT on the other hand is "infinite" dimensional in that it takes an infinite time series as input and outputs a continuous spectrum with period of 2pi.