My main character is a human with grey irises, and I want to know how I could realistically explain her irises glowing - as if the irises themselves are lightbulbs - red or purple during moments of high emotion, and if it would be possible for her to control the glowing.
Glowing is easy, bioluminescence already exists in nature.
Bioluminescence occurs widely among animals, especially in the open sea, including fish, jellyfish, comb jellies, crustaceans, and cephalopod molluscs; in some fungi and bacteria; and in various terrestrial invertebrates including insects. Many, perhaps most, deep-sea animals produce light. Most marine light-emission is in the blue and green light spectrum. However, some loose-jawed fish emit red and infrared light, and the genus Tomopteris emits yellow light.
The most frequently encountered bioluminescent organisms may be the dinoflagellates present in the surface layers of the sea, which are responsible for the sparkling phosphorescence sometimes seen at night in disturbed water. At least eighteen genera exhibit luminosity. A different effect is the thousands of square miles of the ocean which shine with the light produced by bioluminescent bacteria, known as mareel or the milky seas effect.
Non-marine bioluminescence is less widely distributed, the two best-known cases being in fireflies and glow worms. Other invertebrates including insect larvae, annelids and arachnids possess bioluminescent abilities. Some forms of bioluminescence are brighter (or exist only) at night, following a circadian rhythm.
Red or purple eyes, or irises, is a different matter; I don't think that occurs naturally; the closest is probably albinism, but that is more of a lack of pigmentation than otherwise.
From Eye Color-Wikipedia:
The eyes of people with severe forms of albinism may appear red under certain lighting conditions owing to the extremely low quantities of melanin, allowing the blood vessels to show through. In addition, flash photography can sometimes cause a "red-eye effect", in which the very bright light from a flash reflects off the retina, which is abundantly vascular, causing the pupil to appear red in the photograph. Although the deep blue eyes of some people such as Elizabeth Taylor can appear violet at certain times, "true" violet-colored eyes occur only due to albinism.
I'm going to assume you are using a strong definition of the word "realistic".
The crux of the idea is to exploit something along the lines of a tapetum lucidum. Having done a little research (Wikipedia), it seems that the majority of so-called eye-shine phenomena in vertebrates stems from the presence of crystalline inclusion bodies in the cells behind the retina. Colors vary from species to species, so, presumably, with crystals built from the correct proteins, you could get the color you want.
As for making the eye-shine emotion-dependent, I see several possibilities.
The tapetum could be only mildly reflective, weak enough that noticeable eye-shine would only occur when the character's pupils were widely dilated — something that often occurs during moments of intense emotion.
The crystalline inclusion bodies in the cells of the tapetum could be contained in contractile vacuoles. The reflective pigments/crystals would only be exposed when the vacuoles are expanded, which could occur as a simple stress response (triggered by acetylcholine, or whatever).
Funnily enough, the eyes of some people do change when they're feeling certain emotions.
When a person is angry or passionate or very excited (basically whenever a person has high levels of adrenaline within their system) their irises contract to enlargen/dilate their pupils. When this happens the pigments within the irises compress.
This in turn can lead to the eyes of some people to literally change color depending on their mood.
Now while this is true, you would have to explain why her eyes turn purple: a color that human eyes don't normally take.
However, once blue eyes were also a huge oddity. It is believed all blue-eyed individuals share a single common ancestor who gained the trait because he/she was affected by a random genetic mutation.
It is plausible (though very unlikely) that your character would have also experienced one such lucky mutation, this one causing her to possess pigments within her eye that are only very faintly purple. This tint would thus only be visible when the iris is compressed. Otherwise her eyes would indeed seem grey.
Now as for glowing red I don't see how this would be possible.
I have witnessed my exes eyes turn to what seemed like a "glowing" purple - only twice. Both times in EXTREME anger. Like, about to cause serious physical harm anger. I have known him for over a decade now and im not the only one who has witnessed it. His eyes are naturally an extremely bright blue. I think the natural brightness is what caused the "glowing" effect personally.
The glowing can be accomplished by three chemicals, two of which are already present:
1) Glucose (or ATP) as the energy source;
2) Adrenaline (or whatever hormone is in high supply when the emotion in question is in play);
3) A novel protein unknown to real-world medicine. This protein should be expressed only in the places where the glowing effect is desired.
As far as giving the character conscious control over the glowing, I would probably develop a mental discipline which puts the adrenaline levels under conscious control.