Can sinus infection that has moved to hurting above eyes cause swelling in one or both arms?

Sinus infection. Sinus infections are isolated to the head and neck they never involve the lower extremities unless there is a massive infection that involves the veins that drain the head and neck. In these cases the whole head and neck are swollen as the infection extends to the brachio-cephalic veins that drain into the superior vena cava near the heart. There would be other signs and symptoms.
NO. No sinus infection will not cause swelling in one or both arms.

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