Examples where non-linear phase filters are used
Two examples:

Audio: The human ear isn't very sensitive to nonlinear phase (probably because the world filters sound with nonlinear phase filters).  It is, however, sensitive to percussive sounds that "pre-ring"*; i.e., that start making sound before the main "bang".

So for high-fi audio, one often wants to use minimum phase filters (which tend to have nonlinear phase response).  In particular, the use of IIR filters is common (or at least commonly advocated).

Control Systems: In a control system, absolute phase shift (or delay) rather than nonlinear phase shift is the enemy.  FIR filters that have make nice pretty-looking output pulses in response to input pulses make systems that cannot be tuned to the highest possible bandwidth without oscillating -- and systems with pure delay (as you get from a FIR filter) are particularly hard to tune.

So with exceptions (i.e., comb filters can be used to advantage, if you're careful) control engineers want any filtering done with IIR filters, and they tend to want that to be kept to a minimum.

* Or at least there are audio equipment designers that believe this -- I'm parroting things I've read.