What might cause right side stomach muscle to burn and hurt when applying small amounts of pressure or sitting up?

The abdominal muscle is just as prone to injury as any other muscle. Small tears, stretches and strains can occur with exercise, coughing, laughing or any other activity that causes spasmodic clenching of the muscle. Allow the area to rest, use Ice packs and ibuprofen as needed to reduce inflammation, and avoid any exercises for at least 1 week to allow the muscle to heal.

Sounds as though you may have had a soft tissue injury to the skin or musculature in the abdominal wall. Trying not to overdo it, using ice for 20 minute intervals initially, and seeing your doctor if you have worsening symptoms, redness, or abdominal tenderness are all good ideas. Good luck.