Q:

Why didn't Dumbledore's use Fawkes's tears to heal his hand?

After Dumbledore had put on a Horcrux that was also a Hallow, why didn't he use Fawkes's tears to heal his hand?

A:

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.

