On a 2000 toyota celica, the battery is dead for a month, if I jump start the battery and drive it will I hurt the alternator?

How do you know the alternator is good?
You need a digital multimeter to check.
With car off check DC voltage of battery, should be over 12 volts dc.
Start car and measure again.
If alternator is good, dc volts should now be over original reading, if alternator is faulty battery is not charging.
If alternator is good you may have a dead cell in your battery, if so replace.
A give away of a dead cell, is that your original reading of dc volts is under 12V.

Make sure the alternator is producing the rated amperage. One or two blown diodes inside the alternator will reduce its output. Test for a draw. You do this by taking off a battery cable and using a multimeter set to measure current connect it to the battery cable and then to the battery post. Now start removing one fuse at a time while observing the current draw until it stops. Whatever circuit that the fuse controlled is the problem.

The battery connections may be dirty. The alternator may not be charging the battery. You had enough battery to start the car and keep it running but because the battery may not have been receiving charge, it lost power the whole time the engine was running. When you turned off the engine, the battery may have been practically dead. So, it wouldn't have any power left when you tried to restart or even turn on.

If you have a battery charger, charge the battery back up. Check the alternator for proper charging output. some auto parts stores will check the alternator output for free.

What gauge jumper cables are you using? May need a larger set. My mustang wouldn't jump on anything more than 6ga.
Are lights coming on at all? Does the engine make any noise when attempting to start?
Could be an elecreical drain somewhere.
Could be a fuel pump. Does it get fuel?
You need more info with your question.

Sounds like you have a bad alternator, or something drainging your battery, take it to Pep Boys or Autozone, they can do a load test to check the alternator.

Check alternator voltage at back of alternator ( big wire ), if you have no less then 13.5 volt's, alternator is ok, check for blown alternator fuse or break at wire from back of alternator to starter then to battery, could be a bad fuse link at starter.

Sound like you have a bad alternator, poss bad battery too.If you can jump start it with a volt meter check voltage at battery should have above 13.5 volts, do this after you unhook jumper cables and engine running.If bad voltage reading replace alternator make sure you charge battery before starting engine with new alternator or it could damage alternator.If good voltage have battery check.Hope this helped.

you can jump the car and let it warm up and run for awhile to get it warm again it wont hurt it at all

no.the car still draws power from the battery . if you let yor car sit for longe peroids you should disconect battery... your car should be ok if you have any futher promblems let me know

There sounds to be a problem with your alternator. If the car went dead every few hours or week I'd say its the battery. But if the car lasts 3 months between problems, then you're battery is a champ despite fighting a losing battle.  Do you ever take long trips that coincide with this problem? If your battery isn't recharging then it could drain to dead on a longer trip.  There could also be an electrical problem like a bad ground that only shorts bad every once in a while. Read the voltage when the battery is connected to the car with everything off and then again when the battery is disconnected from the car. Also, have you had the alternator checked? Have you inspected the engine for bad grounds? Have you tried buying one of those battery saver devices? Hope this helps.
Paul