Welcome to the site.
Well you see the burning sensation in the hands and feet that you have been experiencing is most likely due to the underlying Grave's disease which results in hyperthyroidism
( overproduction of thyroid hormones) but synthroid
that you have been taking is a drug which is given hypothyroidism
( underproduction of thyroid hormones) so as to increase the thyroid levels in the body, so basically synthroid is a wrong drug to be used in Grave's disease as it is likely to worsen your condition by further increasing thyroid levels which already are above the normal range.
So it is the synthroid which is causing these symptoms as it's use is not justified in this disorder that you have, instead you should have been taking Propylthiouracil
or methamizole drugs for your Grave's disease.
So I will suggest that you first get your thyroid function test done so as to check the levels of FreeT3, Free T4 and TSH
which will give you an idea about how seriously high the levels of thyroid hormones are and accordingly you shall ask your doctor to prescribe you one of these two mentioned drugs which shall take care of your condition and symptoms associated with it.
As for synthroid you shall ask your doctor to discontinue it as soon as possible as it is the culprit and a wrong drug for you in your underlying condition.
I hope this helps.