We do not anticipate damage to the follicles once they are implanted. Post transplant, all patients are instructed on how to care for the grafts and what to anticipate over the course of the next 10 months. Instructions include hair washing, avoiding direct prolonged sun exposure, shock loss, etc.Extreme heat, sun burn and ironing (pulling) hair can damage your native hair too and transplanted hair falls in the same category.
If you follow your doctor's instructions there should be no issues after surgery. Heat from the sun will not damage your transplanted hairs.
The answer is no. Once the grafts are set in place and you do not pick them out and wash your hair appropriately, the grafts will do their job and eventually grow even if you were in the sun. We tell people to avoid direct sunlight to allow the grafts to heal without scarring.