If you push them even farther, there's a 'Shaken' state that kicks in once they drop below 33% Will. If that happens, all it means is they'll have more starting Will if you insist on sending them out anyway, and so be more likely to avoid Panicking and whatnot. This is independent from normal Will regeneration, so even if it would take less time for them to regenerate their Will to max that won't lead to them exiting the Tired state prematurely. Once a soldier has entered the 'tired' state, they'll take 8-12 days to recover from it. This means the Fatigue system doesn't hit Psi Operatives as badly as you might expect, and more generally encourages trying to prioritize sending Tired soldiers to Facility-related stuff so you minimize how many troops are unavailable at any given moment. You're also not allowed to deploy them onto Covert Ops if they're Tired, period, though note that assigning them to Facilities is not blocked by being Tired: they can perform Bond training, Retraining, Psi Operative training, and negative trait removal at the Infirmary just fine.
If it drops below 67%, the soldier will be Tired after the mission ends -you can still deploy them in this state, but it's a very bad idea, as they'll react badly to more or less literally anything bad happening, and once the mission completes they'll probably earn themselves some manner of psychological problem that will impair their performance in future deployments even once they've rested.
Specifically, when you deploy units, their Will 'rots' over the course of a mission. (In part because a lot of elements of XCOM 2 and War of the Chosen are clearly influenced by Long War, and this system is broadly comparable to Long War's Fatigue system) A new mechanic in War of the Chosen that the manual briefly alludes to, but which is not properly explained anywhere in it or the game itself, is what I call a Fatigue system.