I know it sounds weird, but it's slightly reassuring to me as an EMT that your burn hurts. If it didn't, I'd be far more worried, as that would be a full thickness (3rd degree) burn.
For now, just keep it clean. Avoid using antibiotic creams like Neosporin, though, as these are oil based and can make the burn worse. Instead, I'd recommend a disinfecting spray with some lidocaine.
If the burn goes around your finger or covers most of it or if you have any chronic condition that compromises your immune system or could interfere with healing, it might be a good idea to see your doctor.