Covering head with Tallis:

It is the custom of those particular in Mitzvos, and of all the Chassidim[1], to cover their head with the Tallis Gadol throughout the entire period of time that they are wearing it, in order so their head is not uncovered for even one moment.[2] Doing so humbles ones heart and leads to fear of heaven.[3] One who removes his Tallis from his head is like one who passes over the chance to perform a Mitzvah.[4] One is not to uncover his head from the Tallis even if he is very hot.[5] The Tallis is to remain on one’s head from the beginning of Davening until the end.[6] Seemingly, this applies likewise during Kerias Hatorah. Nonetheless, the Rebbe Rashab was not accustomed to cover his head during the Keriah.

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[1] Toras Menachem 18:106

[2] Bach 8; Taz 25:2; Olas Tamid 8:3; Matzos Shmurim p. 23b “The main ourpose of the Tallis Gadol is to cover the head. Unlike those the few Shotim which only garb their body in the Tallis Gadol and not their head, and it is a great mistake in their hands, and they are saying a blessing in vain, as how can they say the blessing of Lihisateif, if they are not garbing their head”; Radbaz 1:343; Shaareiy Teshuvah 8:3; M”B 8:4; Ketzos Hashulchan 7:4; Toras Menachem 18:106 [brought in Shulchan Menachem 11:36]; See Shaareiy Teshuvah 8:3 and Kaf Hachaim 8:9 for different customs in this regard

[3] Bach ibid

[4] Taz ibid

[5] Magen Gibborim 91:3 in name of Meiri; Chavos Yair 53:1

[6] Elya Raba 8:4; Poskim in Kaf Hachaim 8:10; Ketzos Hashulchan ibid