Originally Posted by PatHenry

So let's consider what is going on in general:
1 - When the engine is on, the flywheel is spinning. When it's off, the flywheel is not. The disc grabs onto the flywheel and the pressure plate is bolted to the flywheel - when the clutch is not depressed the transmission is locked to the engine, so the input shaft is spinning at the same rate, when the clutch is pushed in, the transmission is disconnected from the engine, so the input shaft is possibly spinning (if you're moving) but might not be (the vehicle is stopped).
2 - What happens when things heat up? Metal expands. Metal can become more pliable. Fluids can thin. We know that hydraulic fluid looses it's ability to hold pressure when it becomes too hot.
3 - The way the transmission is designed, when the transmission input shaft is spinning at a certain rate of speed, it sort of locks out the ability to shift into a gear that's not consistent with that speed - hence the synchronizing of the input shaft, engine speed, output shaft.
4 - Manual transmissions are extremely difficult to shift if the clutch is not depressed while moving. It is possible, but risks damage to the transmission. They are easily shifted regardless of clutch when the engine is not running.

So let's consider your situation:
1 - Engine running and not a lot of heat present. Shifting is normal.
2 - Engine running and lots of heat present. Shifting is impaired.
3 - Engine stopped with or without heat present. Shifting is normal.

I think it's safe to assume that it's extremely unlikely to be:
- Worn or damaged synchros. This would be something that would cause problems at all times and likely isolated to a subset of gears.
- Issues with the transmission fluid. This would cause issues at all times and make shifting difficult in all conditions.
- Air in the clutch hydraulic line. This would cause problems with all clutch operation regardless of temperature.

So combining the general and specific considerations - it's possible that:
1 - The heat from the engine is causing poor quality or incorrect fluid in the clutch to fail to disengage the disc from the flywheel. That would cause difficulty shifting while moving, but allow shifting when stopped/engine off.
2 - The heat from the engine is causing the fingers on the pressure plate to not press firmly enough to release the disc from the flywheel. Again, that would cause difficulty shifting when moving but allow shifting when stopped/engine off.

So it's most probably an issue with the hydraulics and relationship with disengaging the disc.

If it were me, I'd approach it by:
1 - Seeing about bleeding all the hydraulic fluid out so that new, known correct fluid was in the master/slave system. I don't know how difficult this is as I've never had to do it.
2 - Replace the master/slave system with the full, pre-bled assembly. This is not overly expensive, relatively easy to do and not a ton of time. It is worth it to fire the parts cannon on this one if there's any chance of avoiding #3.
3 - If the problem remains, pull the tranny and put a new clutch set on it. (New pilot bearing, throw-out bearing, disc and pressure plate. I'd also replace the pivot clip for the fork.) This is a lot of work and there are a couple potential trouble spots (upper tranny bolts, aligning the transmission to re-install). It's certainly a job that anyone that is comfortable with turning wrenches can do, but it does require a serviceable floor jack and jack stands, some sort of transmission jack, a good 3+ feet of socket extensions and the inverse torx socket. A slide hammer with pilot bearing puller is helpful and can be borrowed as a loaner tool. Having a helper is also strongly recommended. If you can have the Jeep "in the garage" for at least a weekend (2 weekends and the week in between is even better) then it's a great learning experience and will save a LOT of money over having it done at a shop.