1- It is impermissible to fast on the two days of Eid; Abu Hurayrah t narrated that the Messenger of Allah ﷺ forbade fasting on two days: Day of Adha and day of Fitr.2- It is impermissible to fast the ayyamut- tashreeq, which are the three days after the day of Eidul-Adha [11th, 12th, and 13th of Dhu’l Hijjah]. The Messenger of Allah ﷺ said,
“The days of tashreeq are days of eating and drinking,”
[ Source: Muslim.]
but one performing Hajj may fast these days if he is doing qiraan or tamattu’ Hajj and cannot find a sacrificial animal, as is explained by the Qur’an in Ayah 196 of suratul Baqarah:
“Then if you are in safety and whosoever performs the ‘Umrah in the months of Hajj, before (performing) the Hajj (i.e. Hajj-at-Tamattu’ and Al-Qiran), he must slaughter a Hady (animal, i.e. a sheep, a cow, or a camel, etc.) such as he can afford, but if he cannot afford it, he should observe Saum (fasts) for three days during the Hajj and seven days after his return (to his home), making ten days in all.”
3- It is impermissible to Fast on the yawmul- shak (day of doubt), which is the day that one doubts whether it is the last of Sha’ban or the first of Ramadan, when there is cloud or fog that prevents the sighting of the crescent. Ammar, t said:
“Whoever fasts the day of doubt has disobeyed Abul Qasim (Muhammad, ﷺ).”
[ Source: Tirmidhi.]
source: islamkingdom