Why Doesn’t Kylo Ren Have Sith Yellow Eyes:
Well, first of all, even though it doesn’t have a lot to do with the explanation I should just tell you he isn’t a Sith.
Now, on to the explanation. If this question is bugging you and you’ve been thinking about the meaning of sith eyes or yellow eyes, you have to take a read at a post I made that thoroughly explains Sith Yellow Eyes in general.
Have you read it? Good.
Now we follow through to explain Kylo Ren’s particular case.
If you read the article I recommended, you’ll now know that the degree of corruption and how far the characters have fallen towards the Dark Side of the Force is totally connected to this phenomenon.
As such, you can easily tell that Kylo Ren isn’t that deep into darkness as the plot seems to suggest.
The absence of these yellow eyes is just another evidence that Kylo is redeemable and as such we can speculate that he will eventually turn towards the light as he struggles to keep his heart dark.
You can see in the prequels that Anakin had these eyes at times, because it was his transition, but you can also consider Kylo to be his polar opposite.
No matter how much Kylo loves the idea of Vader, he is quite the contrary to it since Anakin was a light force wielder that struggled to be good, while Kylo is a dark force wielder that struggles to be dark.
This is the simple answer to how Kylo’s eyes remain unchanged, but I suspect he will see intermittent eye colors as he struggles between light and darkness, but only the upcoming films can clear that out for us.