After Dumbledore had put on a Horcrux that was also a Hallow, why didn't he use Fawkes's tears to heal his hand?
Phoenix tears are not a cure all. They are a powerful healing agent for wounded flesh, poison, physical damage and disease. But Dumbledore's hand was not affected by any of these, it was affected by a powerful magical curse that was slowly and steadily spreading. Phoenix tears are not a counter spell. They could not heal a curse.