Sheikh `Abd Allah al-Dumayjî
The hadîth being cited does not lend any support to the idea of Fâtimah’s infallibility. The circumstances in which the statement was made clearly indicate what was intended by it.
Fâtimah’s husband, Alî b. Abî Tâlib, decided to marry another woman as a second wife. When the Prophet (peace be upon him) learned of this, he made this statement, meaning that he was angered by the idea of `Alî marrying the other woman.
These same Shî`ah claim that `Alî himself was infallible – and he is the one who made Fâtimah angry in the first place by his thinking about marrying another woman. She complained about him to the Prophet (peace be upon him), prompting him to say: “Fâtimah is part of me, and whoever angers her has angered me.” [Sahîh al-Bukhârî (3729) and Sahîh Muslim (2449)]
How can we assume that Fâtimah was infallible when the Prophet (peace be upon him) had said to her: “O Fâtimah, daughter of Muhammad, ask of me whatever you wish of my wealth, for I can not avail you anything with Allah.” [Sahîh al-Bukhârî (2753) and Sahîh Muslim (204)]
What could the Prophet’s words to his daughter “…for I can not avail you anything with Allah” possibly mean if Fâtimah had been infallible?
Source: Islam Today