This condition is called anhidrosis and causes muscle cramps, physical weakness, dizziness and more serious problems like heat exhaustion and heat strokes.
Horner syndrome involves miosis or constriction of the pupils, ptosis or the drooping of the eyelid, and anhidrosis which is an inability to sweat in the face.
This leads to Horner's syndrome on the same side of the face as the lesion, which include symptoms of miosis or constricted pupil, ptosis or droopy eyelid, and anhidrosis or failure to sweat.