Well this is good question why does Kakashi Sensei never takes off his mask even after the war, after he became the Hokage also when he retired he still keeps his mask on his face but why so?
Today we will try to find a reason behind this it is all just assumptions as Kishimoto Sensei has not given a particular reason for this but there is something which relates to this that Kishimoto said.
So let’s start.
He Kept it on to hide his emotions
Kakashi has been through a lot of sad moments in his live he lost in mother at a very young age.
He saw his father suicide in front of him, thought that his friend (Obito) died in front of him and had to kill his lover (Rin) with his chidori.
He blamed himself for the loss of both of them for his whole life.
Then even after that he lost his teacher during the nine tails attack.
He is a character who in his early shinobi days was strict on following the ninja codes and one such rule was that a ninja don’t have any emotions he is just a tool for his village.
So maybe having that mask on his face was one of his way to cope up with all those emotions, hide his tears from anyone to see.
Kishimoto Reason Behind it
Kishimoto Sensei the creator of Naruto had said that in the beginning he had planned to create all the shinobis wearing a mask as this was necessary to hide the emotions of ninja.
But then he thought that without the emotions he would not be able to convey the depth and stories feeling without including them so later he kept some character with mask and some without it.
He kept Kakashi wearing his mask as his characters design and personality gets complemented by the mask and it make him look cool.
Danger From Enemy villages
Another reason of him to keep that mask on could be because of his father the White Fang Sakumo Hatake role in being the most skilled assassin of Konoha and even comparable to be above the sannin.
There was great risk on Kakashi who has a very similar face to his father which stayed like that even in his old days.
Many shinobi, the tsuchikage and even the third daimiyo of fire nation were able to know he is Son of Sakumo from Konoha from seeing his looks.
And there would been much danger to his life atleast in his childhood and then it could have became a habit of him.
He Don’t Have A Face
And the most interesting reason behind him wearing that mask everytime is because he don’t actually have a face.
Listen what I meant by this is that Kishimoto Sensei never came up with the idea of how he would look without a mask and the face we have seen in anime is non-cannon.
So maybe Kishimoto Sensei has never decided how his face should look like hence the face mask.
Now the only thing remaining on his face is his other eye patch so the reason behind him covering his other eye with the Sharingan is that it continuously consumes his chakra the more he left it open.
And he don’t come from Uchiha or Uzumaki clan who has immense chakra reserves.
So to conserve his chakra he keeps his eye with the Sharingan covered most of the time.
And he stopped covering it in Boruto as now he no longer has the Sharingan but the above causes could still apply for covering his face.