Burning on right side of penis mostly after sex. No redness or swelling. Only one partner. What can this be?

Abrasion. Sounds like abrasion injury to the shaft of your penis. Often you might get a raw appearance in the sore area, but not always. Allowing your partner to be fully ready and lubricated should help prevent these symptoms, as well as using a lubricant such as K-Y jelly. Hope that helps.
Get it checked. A burning penis needs to be examined . It could be an infection, of the penis, the urinary tract and much more. Only a complete history and exam will give you the best diagnosis. Please see your doctor.
Check up. Inspect to see wether you have a minute lesion.An abrasion,apimple.If that fails to reveal anything that might explain,you definitely need to be evaluated by your doctor.