Eyes change color to yellow with deep immersion in Dark side, but exact mechanism is unknown
It is said that Dark Side represents destruction, corruption, will to power and egocentrism. Palpatine was almost perfect embodiment of these Dark Side ideals, so with his constant immersion in Dark Side, when not wearing his Chancellor mask, his face was disfigured and his eyes yellow.
On the other hand, his apprentice Dooku and assassin Ventress do not have yellow eyes. Dooku was never fully egocentric like Palpatine, he has his ideals about "Human empire" and that ultimately cost him his life. Also, Dooku tries to keep his gentlemanly demeanor even when he does some truly despicable acts. Ventress is a similar case, there is constant fight in her soul between good and evil, finally she chooses Light. It seems that both of these characters never fully embraced darkness, perhaps because their prior training as Jedi.
Anakin is most curious in this regard, his eyes constantly change color. Interestingly, when he does his most gruesome act - killing of younglings, his eyes are normally blue
but when left burning on Mustafar, his eyes are burning yellow. This could probably mean that he used all of his hatred and Dark Side power to help him survive (just like Maul did in similar circumstances), but he didn't need that to help him murder children .
Maul is another interesting example, his eyes are constantly yellow, even after his death in Kenobi's arms, when he supposedly let go off his hate. Note that Maul is actually only true student of Palpatine, raised by him since childhood unlike Dooku and Anakin who were once Jedi. Perhaps he picked up too much from his old master and could not shake it off even if he tried. I don't think his natural eye color was yellow, and reason is his brother Savage.
Savage Opress had yellow eyes while he was alive and enraged, but in his last moments they returned to natural green-blue color. This happened when Magick cast on him withdrew, implying that maybe his brother Maul had same or similar eye color before Palpatine got him as an apprentice.
Finally, we have Snoke. Although Disney really fumbled with this character, there are few things about him that lead to certain conclusions. Snoke is not a Sith, but he is very powerful with Dark Side, comparable to Palpatine (some say even more powerful) . Yet, his eyes are blue. Does this mean he is not fully immersed in darkness? Possibly, because he did mumble something about darkness rising and light to meet it, implying that Snoke believes in balance rather then just one side of the Force. Perhaps this is the reason he tries to connect Kylo and Rey? Anyway, this is all done very poorly, so we leave it at that.