Pupils dilate in dark areas to improve vision. In a well-lit area, the pupils will be small because there is a lot of light reaching the eye. They gets bigger in darker places to allow more light to enter the eye. That's why when you walk outside after being in a really dark room, it takes a while for your eyes to adjust to the increased lighting.

*science nerd*
I'm not saying this is the reason for Kovu's weird looking eyes, but keep in mind that the fire scene was extremely bright. Of course that doesn't explain why Kiara's pupils were still really large, but I guess his eyes was just more sensitive to it lol

In the dark scenes, such as the pic OP linked to, his eyes are "normal" sized.
As someone else pointed out, it also makes him look more shocked/terrified. Simba's pupils were pretty tiny too when he watched his father tumble down the gorge:
