More points:
1) The HDMI input on some (most?) 120/144Hz monitors may only work at 60Hz anyway.
2) If you're thinking HDTV's with 120Hz or 240Hz options for smoothing motion then absolutely do NOT turn that feature on. It samples several frames then artificially creates new ones which adds a lot of input latency. It's still a 60Hz input.
3) As said, no advantage, plus I'm not sure if the PS4 will even have a 120Hz or 144Hz option during setup anyway. I suspect 1080p@60Hz would be the highest supported.