In the name of Allah, the Most Gracious, the Most Merciful. O Allah, bless Muhammad and the family of Muhammad.
The threat of Umar ibn Al-Khattab to burn the house of the daughter of the Prophet, Fatimah Al-Zahra (peace be upon them). Here are evidences from the narrations, hadiths, and books of Ahl al-Sunnah:
Aslam Al-Qurashi, a freed slave of Umar ibn Al-Khattab, narrated: "When Abu Bakr was pledged allegiance to after the Prophet (peace be upon him), Ali and Zubair used to visit Fatimah, the daughter of the Prophet, and consult with her about their affairs. When Umar ibn Al-Khattab learned of this, he went to Fatimah and said: 'O daughter of the Prophet (peace be upon him), by Allah, there is no one dearer to us than your father, and after your father, no one is dearer to us than you. By Allah, that will not prevent me, if these people gather with you, from ordering that they burn your house.' When Umar left, they (Ali and Zubair) came to her, and she said: 'Do you know that Umar came to me and swore by Allah that if you return, he will burn down your house? By Allah, he will certainly do what he has sworn. So depart, guided by righteousness, and do not return to me.' So they left her and did not return until they had pledged allegiance to Abu Bakr."
This narration is recorded by Ahmad in "Virtues of the Companions" (1/364), Ibn Abi Shaybah in "Al-Musannaf" (7/432), and Ibn Abi Asim in "Al-Mudhakkir wal-Tadhkir" (1/91). It is also narrated by Ibn Abd al-Barr in "Al-Isti'ab" (3/975) through Al-Bazzar, although I could not find it in the printed books of Al-Bazzar. Al-Khatib also mentioned it in "Tarikh Baghdad" (6/75) in a shortened version: all of them narrated it through Muhammad ibn Bishr, from Ubaydullah ibn Umar, from Zayd ibn Aslam, from his father.
I say: This is an authentic chain of narration, as Muhammad ibn Bishr Al-Abdi (d. 203 AH) is a trustworthy memorizer, included in the six major books of hadith. Similarly, Ubaydullah ibn Umar Al-Umari, who died in 140 AH, and Zayd ibn Aslam, a freed slave of Umar ibn Al-Khattab (d. 136 AH), and his father Aslam, who, according to "Tahdhib al-Tahdhib" (1266), lived during the time of the Prophet (peace be upon him), though he was not in Medina during the events of the pledge of allegiance, as mentioned by Muhammad ibn Ishaq: "Abu Bakr sent Umar in the year 11 AH to perform Hajj for the people, and he bought Aslam, his freed slave, at that time." Thus, the hadith is considered mursal (interrupted), though it is likely that Aslam heard the story from Umar ibn Al-Khattab or other companions who lived through those events.
Here are some of the reports narrated by esteemed Sunni scholars about the burning of the house of the Prophet's daughter, Fatimah, by Umar ibn Al-Khattab:
From Al-Baladhuri in "Ansab al-Ashraf" (Volume 1, Page 586): "Al-Mada'ini, from Maslama ibn Muharib, from Sulayman al-Taymi, from Ibn Awn: Abu Bakr sent to Ali demanding allegiance, but he did not give it. Umar then came with a torch. Fatimah met him at the door and said: 'O son of Al-Khattab, are you going to burn my door?' He said: 'Yes, and that will be stronger in what your father has brought.' So Ali came and pledged allegiance."
Abdullah ibn Muhammad ibn Abi Shaybah Al-Kufi Al-Absi, who died in 235 AH, narrated in his book "Al-Musannaf" in the second volume in the chapter "What Came About the Caliphate of Abu Bakr and His Handling of Apostasy," saying: "Muhammad ibn Bishr narrated to us, Ubaydullah ibn Umar narrated to us, Zayd ibn Aslam narrated to us from his father Aslam: 'When Abu Bakr was pledged allegiance after the Prophet (peace be upon him), Ali and Zubair used to visit Fatimah, the daughter of the Prophet (peace be upon him), and consult with her about their affairs. When Umar ibn Al-Khattab learned of this, he went to Fatimah and said: 'O daughter of the Prophet (peace be upon him), by Allah, there is no one dearer to us than your father, and after your father, no one is dearer to us than you. By Allah, that will not prevent me, if these people gather with you, from ordering that they burn your house.' When Umar left, they came to her, and she said: 'Do you know that Umar came to me and swore by Allah that if you return, he will burn down your house? By Allah, he will certainly do what he has sworn. So depart, guided by righteousness, and do not return to me.' So they left her and did not return until they had pledged allegiance to Abu Bakr."