I've been told - though this is secondhand info - that if the burn damage is extensive enough, temperature regulation can be messed up.
Only during the acute phase of the burn, typically. Once healing has occured, this usually isn't a problem.
The biggest problem with burns is infection, which can lead to sepsis and death. However, for the OP's character, the biggest challenge is going to be eschar and contractures. As the skin heals, it tightens and can pull extremities into twisted shapes, and the person loses full range of motion over the affected area. This can result in distortions in the face as well, or the ability to stand up straight.
Surgery to release the tightened skin helps this (escharotomies). Scarring in such cases is usually pretty extensive, and takes a long time to heal. This character likely has to use lotions and other products to soften the skin, and do frequent range of motion exercises to maintain mobility.
Some problems the character would face would include body image disturbances (talking in Hero System terms, a Distinctive Feature Disadvantage), the contractures would represent a Physical Limitation, anxiety over appearance a Pysch Lim, and the use of lotions for the skin a Quirk (in GURPS terms). The lotion might also leave the person with a distinctive smell (a Quirk or possibly Distinctive Feature).
Chronic pain might also be another issue to deal with. Also, weakness of an extremity depending on the severity of the burn.