hello CTLE is a continuous time filter, so don't cal it as FIR or IIR. DFE is an nonlinear means it has sampler in between, so even though it is digital it wont come under FIR and IIR. Mainly ctle is to equalize all kinds of cursors but DFE is mainly to compensate post cursor ISI without amplifying the noise where all other equalizer amplify the noise in general,  crosstalk to be specific, but the problem with DFE is it can work in the links where the BER is already high (around 10^-10), otherwise it add more ISI in opposite to the reduction. This is called error propagation.

Please refer this website.

http://www.ece.tamu.edu/~spalermo/ecen689.html

who said it is FIR....check slide 18 here http://www.ece.tamu.edu/~spalermo/ecen689/lecture8_ee689_rxeq.pdf


he is saying that DFE is composition of FIR and sampler.  In fact now people are trying to implement with IIR in place of FIR