It's not surprising to come away from a session of boxing with sore forearms. Given the impact involved, especially in contrast to the relatively small bones and muscles in your forearms, the surprise is that the soreness isn't far more frequent and severe. It's just part of the sport of boxing, though you can take steps to alleviate the discomfort and watch for signs of serious injury.
Simple Muscle Soreness
Muscle soreness in your forearms from boxing has the same cause as most other kinds of muscle soreness. When you repeatedly hit a target with the force involved in a boxing punch, the muscles in your forearms sustain microscopic tears. Those tears heal stronger and thicker -- this is the basic physiological mechanic behind building muscle -- but they will ache for a few days after your workout.
If you ice your forearms for 20 or 30 minutes after a particularly strenuous boxing session, this will cut down on the inflammation that causes much of the discomfort. An over-the-counter anti-inflammatory pain killer like ibuprofin can also reduce the swelling and ease the pain.
You can't entirely prevent forearm soreness from boxing, but you can take steps to minimize the discomfort. The first rule of thumb is to not clench your fist before you strike. Keep your hand loose even on it's way to the target, and relax it immediately afterwards, in order to minimize forearm soreness. In light to moderate punching sessions, box with your gloves on but without hand wraps. This forces your forearm muscles to work harder in stabilizing your wrists, strengthening the muscles so they're more resistant to the impact. In heavy workouts, wrap your hands and wrists to brace them. The wraps will help protect your forearms from the increased impact of that workout. Contact with the bag or mitts should occur at the first two knuckles of the fist with the hand completely pronated.
Dull aching in your forearms is an unavoidable part of boxing. However, sharp pain can be a sign of a more serious injury; as can pain or limited range of motion in your wrist or elbow, or any pain that feels like it radiates or shoots along your arm. Numbness is another bad sign. If you experience any of these symptoms, you should check in with your doctor as soon as possible. Left untreated, these serious injuries could leave you on the bench for weeks or months -- or ultimately become an injury that ends your fighting days altogether.