Can Applying Heat on Closed Eyelid After Botox Help?

Dear PKeye
What you are describing is a complication of BOTOX treatment. The BOTOX has seeped into the eyelid and weakened the muscle that raises the upper eyelid. This condition is called eyelid ptosis. It represents a pretty severe problem that is fortunately going to wear off. The problem is how disable will you be while waiting for the medication to wear off.
There are medication that may help open the eyes. The most commonly used product is called iopodine. It is used twice a day. Ask you treating physician to prescribe this medication for you. If you were not treated by a physician, see an ophthalmologist or eye plastic surgeon (ASOPRS.org has a directory by area).
Upper eyelid ptosis following BOTOX is no an all or nothing effect. There are different degrees to which the medication can seep into the eyelid. When it is bad, the effect can last 6 or more months. When it is mild the effect may only last a few weeks. Your first week with the Iopodine drops will help determine how long your lid will remain droopy. If you respond to the Iopodine in the first week you start using it, then it is likely that the effect will last about 4 to 6 weeks. If there is not initial effect of the iopodine, then it is likely that the ptosis will last 4 to 6 months and possibly longer.

 Applying heat to the Botox area will not help.  Only time will help as the Botox needs to be metabolized by the body.  There are drops that your doctor can prescribe for you that will help lift your eyelids 1 to 2 mm.  Good luck.

Heat is unlikely to help or expedite the resolution of this condition.
Time is your best friend.
You may want to consider the use of lopidine (apraclonidine) or alphagan (brimonidine) may produce a 1-3mm elevation of the lid.