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Hi there. Burning sensations, wherever they may be, are what we call paresthesias, and typically indicate nerve irritation or nerve injury. In the vast majority of cases, the problem is from repetitive microtrauma to muscles, tendons or ligaments from overuse. In other words, it is related to the things we do over and over, and the irritation (and inflammation) build up very slowly over time.
There may be acute injury (strain from lifting) on top of chronic overuse. Either way, the muscle fibers and/or tendons are usually the structures that are injured on a microscopic level. The nerves are firing either because they are also injured within the muscles or tendons, or are simply trying to let you know that the structures they lie in (and are "responsible for") are being injured.
Of course, and the best way to care for this type of problem is rest, rest and more rest. Wear wrist splints (both to unload the forearm muscles and to let people around you know there is a problem) and try to avoid repetitive tasks. Even using the computer keyboard and mouse is a common source for this type of forearm strain and/or tendonitis. Other than that, some get relief from icing the affected areas (reducing inflammation) and from anti-inflammatory medications such as advil or alleve.