Covering all of the head in the state of Ihram is Haraam for Muhrim men, and the obligatory precaution is that they do not cover even a part of the head. But covering the head with hands or with a pillow during sleep is permitted. Also putting the strap of bag on the head is permitted. Covering the face is permitted for men.
It is an obligatory precaution not to cover the ears, too.
Tying a handkerchief around the head because of headache has no problem, also drying the head with handkerchief or towel is permitted on condition that the pilgrim does not cover the head with that.
Whenever a Muhrim forgets and covers the head, it is Mustahab to say Labbayk again but it is not obligatory.
It is an obligatory precaution not to cover the head with “mud” or “henna” etc. or putting something like a basket on the head.
A Muhrim is not permitted to sink his/her head into the water, even if the rest of the body is out of the water, but pouring water on the head for Ghusl or other purposes and taking a shower is permitted; but a Muhrim should not sink his head into the water in the bathtub.
The atonement of covering the head is one sheep for men as an obligatory precaution, but in case of ignorance or forgetfulness there is no atonement.