My stomach feels like it's burning. Does this mean that I need to eat more?

No. May be gerd or ulcer need to see gi.
No. This could be a symptom of an acid related disorder such as gastriis or esophageal reflux. A 2 week trial of Prilosec otc and a visit to your doctor afterward would be reasonable.
Poor digestion. When the food is not digested becomes putrid and creates Lactic Acid which burns. So start taking bromelain with betaine hcl and slowly increase betaine hydrochloride until the burning becomes better. Roughly for people under 150 pounds will need 1000mg of betaine and 1000 more for each 100 pounds. If pain does not resolve in 2 weeks see a md.