Urg! So - I've just started to really get involved with my puppeteering. It's not necessarily a full time job but - through contacts I've made teaching acting and improv to young students (my puppets came in to play a LOT here - it got to a point where I was making a new one nearly weekly for the kids to have specific characters to choose from. Anyone ever make a Batman muppet-style puppet? Ridiculously difficult. Anyway...) So - with the frequency of my performing I've started to notice a horrible pain in my forearm (the "mouth arm" after I finish.  Anyone have any good exercises to do to strengthen that muscle? Or does it get stronger with practice?

You have said the one key word... Practice.   Pretty much the common reply given to this question. The position and angle used to puppeteer is using a fairly unique set of muscles so practice is going to do you the most good. Not to mention why waste time doing some exercise when you could be getting better at your manipulation.

I should point out though that you said "horrible pain". Muscle ache is one thing but if you have constant pain then you might want to check with your doctor.

I have arthritis and as I got older I had to find other ways to keep doing puppets.  I am surprised your forarm hurts, what position to you use to hold up your puppet.  When I was on my knees it was my upper arm that got tired first.  When I went to lower bent arm puppetry we made a shelf with pillow to rest our elbows on, we could work much longer that way when we could rest a few moments on the shelf. Then it was the back of my hand that hurt by the third show in a row.  Are you leaning your arm out at an angle, if so maybe holding it more upright might be less stressful on forearm. Hope you find way to not hurt so much.

Muscle ache is normal, but pain (especially sharp pain!) is not. Make sure you always perform with your muscles warmed up, loose and relaxed. Being too stiff or rigid can sometimes cause injury. If you're seriously involved in puppetry it is a really good idea to get to know a good physio therapist. They can give you exercises that will help and can suggest strategies to make sure you don't injure yourself. 

Yes, I agree with Andrew.  You could be developing carpol tunnel or tennis elbow or something.  A good physical therapist with one visit could help. As Shawn also said, you are using muscles that most folks would not be.  Other than videos I'm not sure I'd be able to do lengthy live performance.