It could be repetitive motion injury (sometimes called carpal tunnel injury). It might be from too much signing or signing incorrectly. It also might be a cumulative result of signing, texting and some other repetitive motion that you do at work.
Try to make note of when exactly the pain starts, where it's located, and how long it lasts. If it continues, you might need to make a doctor's appointment.
Some people (usually older) require surgery but you probably just need to make some changes in how you do things. That might require occupational therapy.
You can try warming up exercises for your hands and arms before signing. Also, make sure that you are relaxing when you sign--don't tense up. Make sure you are forming your handshapes correctly. You'd be surprised how many people contort their fingers and hands unnecessarily trying to make an incorrect "E," or face their signs in the wrong direction.
One weird pain I used to experience when interpreting for college classes was the nerve in the palm of my non-dominant hand that would shoot up my arm. It happened because I was using more than normal instances of signs that pressed on that palm nerve, such as OWE, DEBT, SHOW, EMPHASIZE, IMPRESS, REQUIRE, MEDICINE. It became so sensitive I could barely touch it. I had to modify how I did those signs.
As you become more fluent in signing, you'll be able to develop economy of movement. That is, when you sign, as you transition from one sign to the next, or fingerspelling one letter to the next, you will flow more smoothly. That means, you don't actually make as many movements, or you make abbreviated ones.
