Praise be to Allaah.
Fasting on the two days of Eid is haraam, as indicated by the hadeeth of Abu Sa’eed al-Khudri (may Allaah be pleased with him) who said: The Prophet (peace and blessings of Allaah be upon him) forbade fasting on the day of al-Fitr and al-Nahr.” Narrated by al-Bukhaari, 1992; Muslim, 827. The scholars are unanimously agreed that fasting on these two days is haraam.
It is also haraam to fast on the days of al-Tashreeq which are the three days after Eid al-Adha (the 11th, 12th and 13th of Dhu’l-Hijjah) because the Prophet (peace and blessings of Allaah be upon him) said: “The days of al-Tashreeq are days of eating, drinking and remembering Allaah.” Narrated by Muslim, 1141.
Abu Dawood (2418) narrated from Abu Murrah the freed slave of Umm Haani that he entered with ‘Abd-Allaah ibn ‘Amr upon his father ‘Amr ibn al-‘Aas. He offered them food and said, “Eat.” He said, “I am fasting.” ‘Amr said: “Eat, for these are the days on which the Messenger of Allaah (peace and blessings of Allaah be upon him) would tell us not to fast and forbade us to fast, and these are the days of al-Tashreeq.” Classed as saheeh by al-Albaani in Saheeh Abi Dawood.
But it is permissible to fast on the days of al-Tashreeq for a pilgrim who does not have a hadiy (sacrificial animal). It was narrated that ‘Aa’ishah and Ibn ‘Umar (may Allaah be pleased with them) said: No one was allowed to fast on the days of al-Tashreeq except one who could not find a hadiy. Narrated by al-Bukhaari, 1998.
Shaykh Ibn ‘Uthaymeen said: It is permissible for those who are doing Hajj qiraan or tamattu’, if they cannot find a hadiy, to fast these three days so that the Hajj season will not end before they fast. But apart from that it is not permissible to fast on these days, even if a person is obliged to fast for two consecutive months; he should not fast on the day of Eid and the three days following it, but then he should continue his fast.
Fataawa Ramadaan, p. 727.
See also questions no. 21049 and 36950. And Allaah knows best.