Are there any other gods that permanently had an eye removed, other than Odin?
I can't find any other gods that had one of their eyes other than Horus, which wasn't even permanent.
Looked here already:
It wasn't permanent, but I did find a similar story. In Hittite mythology, the sky/storm god Tarhunt had as his nemesis a snake/dragon god Illuyanka. Note the similarities here between the story of Thor and the World Serpent. Both are Indo-European peoples, so this is likely not coincidental.
There are two surviving accounts of this story, both from CTH 321. In the second*, Illuyanka wins an initial fight with Tarhunt, and takes his heart and eyes. So now blind, Tarhunt has to rely on trickery. He marries, fathers a son, and then marries him off to the snake's daughter. With that entry to the snake's home, Tarhunt gets his son to retrieve the eyes and heart so that he may renew the battle (at which Tarhunt is victorious and kills the serpent, but at the cost of his own son).
(21) The serpent defeated the Storm-god and took his heart and eyes. ...
(22) And he took as his wife the daughter of a poor man, and he sired a
son. When he grew up, he took as his wife the daughter of the serpent.
(23) The Storm-god instructed his son: "When you go to the house of your
wife, then demand from them my heart and eyes!"
(25) When he was again sound in body as of old, then he went once more to
the sea for battle. When he gave battle to him and was beginning to
smite the serpent, then the son of the Storm-god was with the serpent
and shouted up to heaven, to his father:
(26) "Include me - do not show me any mercy!" Then the Storm-god killed the
serpent and his own son.
So while he did get eventually his eyes back, it took quite a while, and he had to sacrifice his own son to do it.
* - Interestingly, in the first story, killing the serpent involves tricking it into getting tied up, much like the Norse story of Loki's other monstrous offspring, Fenrir.
Perhaps the Graeae bear mentioning. The Graeae were 2-4 sisters who only had one eye (and one tooth) to share between them. There is not any indication that means there was ever a loss of another 3-7 eyes.
However, Perseus stole their sole eye on his way to slay Medusa. In the most well-known version of the myth today, the eye is then ransomed back to them for information. However, is other versions of the myth, the eye was not returned, and Perseus merely took advantage of their blindness to pass by them:
According to Hyginus Astronomica, the eye was not returned, but rather was cast into a nearby lake:
They are thought to have had but one eye among them, and thus to have kept guard, watch one taking it in her turn. This eye Perseus snatches, as one was passing it to another, and threw is in Lake Tritonis. So, when the guards were blinded, he easily killed the Gorgon when she was overcome with sleep.
And similar according to Ovid's Metamorphoses, while no mention of what became of the eye is given, it certainly doesn't seem to be returned to them:
There is a spot beneath cold Atlas, where in bulwarks of enormous strength, to guard its rocky entrance, dwelt two sisters, born of Phorcys. These were wont to share in turn a single eye between them: this by craft I got possession of, when one essayed to hand it to the other.—I put forth my hand and took it as it passed between: then, far, remote, through rocky pathless crags, over wild hills that bristled with great woods, I thence arrived to where the Gorgon dwelt.
It seems that the Graeae were almost certainly immortal (2/3 of their other siblings, the gorgons, were, anyway), but whether they fit your definition of "gods" I can only leave to your judgement.