LCDs themselves exhibit surprisingly little latency, especially TN designs (which do have other problems). VA designs are generally the worst as they require non-causal overdrive waveforms to get decent response time (which is a separate matter), and of course the universe is causal, so things generally just get delayed to make it work. Raw parallel interface TN panels generally have sub-frame latency (usually they latch at the end of a line, it seems), and raw IPS panels are pretty close.
What kills is the scaler and other processing pipeline latency. Devices marketed as "televisions" tend to be much worse in this regard than devices marketed as "monitors" since they're generally trying to do various tricks to improve the perceived image quality of marginal (overcompressed, repeatedly scaled, etc.) photorealistic scenes while monitors are intended to simply display exactly what they're fed since it's usually uncompressed line art (PC graphics).
Some TVs do have a, usually poorly documented, way to turn off a lot of the processing. It's often called "game mode", or, on Samsungs especially, telling it that an input is a "PC" will do it. They're still usually higher latency than a real "monitor".
As to whether you'll notice, it depends heavily on what you're doing with it and how bad the latency is. 1 60Hz progressive frame is almost unnoticeable except in controlled test situations. 2 frames, or a single 30Hz frame, is sometimes noticeable in high-motion, timing-sensitive applications like technical fighters (e.g. Street Fighter), Japanese music/rhythm games (American titles are usually quite forgiving), "bullet hell" shmups, etc. 3+ starts to become noticeable with regular interaction making e.g. interactive aiming in slow-pace shmups hard. 4+ pretty much rules out anything but casual gameplay.
Most humans absolutely have the ability to perceive 1-2 frames of lag in controlled situations, but most won't notice it, even if trying to, in normal applications.
FWIW, I've done tests since I play a lot of Japanese rhythm/music games. beatmaniaIIDX, in particular, is highly sensitive to this particular metric. I can play without meaningful adjustment (i.e. same as a lag-free CRT) with 1 60Hz frame of lag, but 2+ requires conscious adjustment, and 4+ starts to become essentially unplayable without using the hidden mod. Players of that game who use sudden+ mode (aka "towel mode") often have to adjust their setpoint even for fractional-frame lag.