Setting the home of Lady Zahra on fire by Umar and the usurpation of the Fadak by Abu Bakr were both things that took place only a little after the passing of the prophet (pbuh), but as to which one preceded the other, we must say that in Sunni history sources, it has been said regarding the burning of Lady Zahra’s home that: “... ثم إن أبا بکر بعث عمر بن الخطاب إِلى علی و من معه لیخرجهم من بیت فاطمة رضی الله عنها، وقال: إِن أبوا علیک فقاتلهم. فأقبل عمر بشیء من نار على أن یضرم الدار، فلقیته فاطمة رضی الله عنها و قالت: إِلى أین یا ابن الخطاب؟ أجئت لتحرق دارنا؟ قال: نعم، أو تدخلوا فیما دخل فیه الأمة....”  , meaning: Abu Bakr sent Umar to Ali and those with him to bring them out of the home of Fatimah (as), giving him orders to fight them if they were to oppose and resist! Umar, while carrying fire with him, headed for Lady Zahra’s home to set it on fire! He then encountered Fatimah (as). She asked him: “Where are you going? Do you intend on setting my house on fire?!” Umar said: “Yes, unless you accept what the people have accepted (the caliphate of Abu Bakr)…”.
Also, regarding the time in which the Fadak was usurped, it has been recorded that: “عَنْ حَمَّادِ بْنِ عُثْمَانَ، عَنْ أَبِی عَبْدِ اللَّهِ عَلَیْهِ السَّلَامُ قَالَ: لَمَّا بُویِعَ أَبُو بَکْرٍ وَ اسْتَقَامَ لَهُ الْأَمْرُ عَلَى جَمِیعِ الْمُهَاجِرِینَ وَ الْأَنْصَارِ، بَعَثَ إِلَى فَدَکَ مَنْ أَخْرَجَ وَکِیلَ فَاطِمَةَ بِنْتِ رَسُولِ اللّه مِنْهَا”  , meaning that: “When the people did bay’ah (pledged him their allegiance) with Abu Bakr and he took power over all the Muhajirin and Ansar, he sent someone to the Fadak to have her agent expelled from there.”
This passage implies that the usurpation of the Fadak took place several days after the burning of Lady Fatimah’s (as) home. For information on the usurpation of the Fadak, see:
Question 5358 (website: 5580)
Returning the Fadak to the children of Imam Ali (as) during his rule.
 Abul-Fida’, Al-Mokhtasar fi Akhbaril-Bashar (The Tarikh of Abul-Fada), vol. 1, pg. 107, the chapter on the akhbar of Abu Bakr the Siddiq and his caliphate; Ibn Abd Rabbih, Aqdul-Farid, vol. 2, pg. 7. (These addresses have been adopted from the Al-Maktabatul-Shamilah software).
 Biharul-Anwar, vol. 29, pg. 127, hadith 27 (from the Mo’jame Mozu’iye Biharul-Anwar software).