Sunburn or Heat-Related Injury: The burning sensation in the left forearm that worsens with sun exposure strongly suggests a condition related to sun exposure. The absence of chest pain (cp) or shortness of breath (sob) further supports this diagnosis, as it rules out more severe cardiac or respiratory issues.

Neuropathy: Conditions affecting the nerves, such as neuropathy, could cause burning sensations in the arm. However, the specific worsening with sun exposure is less typical for neuropathy.
Dermatological Conditions: Certain skin conditions like eczema, dermatitis, or allergic reactions could cause burning sensations and might be exacerbated by sun exposure.
Musculoskeletal Injury: Overuse or strain of the forearm muscles could lead to pain and burning sensations, potentially worsened by activities that involve sun exposure, like outdoor work or sports.

Deep Vein Thrombosis (DVT): Although less likely given the symptoms, DVT can cause arm pain and swelling. The risk of pulmonary embolism makes it a critical diagnosis not to miss, despite the low likelihood based on the provided symptoms.
Cardiac Ischemia: While the patient denies chest pain, atypical presentations of cardiac ischemia can occur, especially in diabetic or female patients. The absence of sob does not entirely rule out cardiac issues, especially if the patient has a high-risk profile.

Complex Regional Pain Syndrome (CRPS): A chronic pain condition that can cause burning sensations, usually after an injury. It's less likely given the acute onset and specific trigger of sun exposure.
Erythromelalgia: A rare disorder characterized by burning pain, heat, and redness in the hands and feet, and sometimes arms. It could be considered if other diagnoses are ruled out, though it's uncommon and the presentation might not fully match the described symptoms.