Jeep Grand Cherokee Brake Light keeps burning out

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you have a short in one of the light sockets or a worn wire from the wiring harness. could be the flasher assembly. are the turn signals different bulbs from the brake lights? Does using the brakes blow the fuse? (Rear lights or brake light switch on brake pedal). left or right turn signal blowing the fuse will tell which side has the short. 4 ways only, replace the relay.

The most likely problem is two burned out light bulbs. A burned out tail/stop lamp on the right side and a burned out brake lamp on the left. If you have already replaced the bulbs, then yes, the circuit boards are the next most likely cause of the problem.

When you install the new bulbs, are you applying di-electric gel? It keeps moisture out, which I think is the problem

the problem is with the bulb, what is happening is when you push your brake pedal with the lights off you are turning on you parking light side of you bulb, when the lights are on the parking light is already lit up so when you push the pedal you see no change. try a new bulb

Well. The rear brake light go out is a command problem for the 99-04 Jeep WJ. I personally think there is a short somewhere from the factory design.

Remove screws for lens. You may have to access them from inside the hatch. Then pull out lens, then turn and pull out bulb and socket, Push and turn then pull bulb out.

you need to find the intermittent short.

YES -- process of elimination. Obviously there's a short circuit in the Brake/Headlight circuits - right? Blown fuse(s) is/are evident of this -- would you agree? Which fuse(s) is/are blowing? An analysis tells you that the only time a fuse blows is when the headlamps are turned on -- correct? Otherwise, brake lights work normally -- right? OK. Head lamps have relays which turn them on. Here's a wiring diagram of the 2000 Hyundai Sonata (NOT the Jeep Grand Cherokee)
It would appear the problem is with one or more bulbs or sockets or both. As you many notice, there are no connections between the headlamps and brake lamps except the parking lamps (which share the bulbs/sockets)
When the headlamps are turned on, so does the tail lamp relay. As you can see, these bulbs/sockets are 'DUAL' filament to ground (series-parallel circuit). Just a guess here - the short is in one of the lower rear brake/parking lamp bulbs/sockets.... just a guess.

check the wiring and sockets for each bulb in the circuit