My dad is more than 60 years old, he has hypertension and has developed a pain "like burning" he describes, on his both feet, what could be the cause?

Quench the burn. The burning symptoms, especially on both feet, could be the beginning signs of a bigger problem. No need to wait and see. Have him examined by a family doctor or Internist. I find it is best to address these symptoms early and prevent potential issues later.
He may . Have neuropathy. Is hypertension the only issue? What about any history of back pain? .