Is there any plausible reason for a robot's eyes to glow?

There's very little reason to have glowing eyes in the visible spectrum. However, in the IR spectrum, we do see them:

The idea is that most IR cameras are not as sensitive as the human eye, so they need extra illumination to see anything. Putting the IR illumination right next to the camera proves to be very effective -- there are no shadows in the scene when you do this, so there's nowhere to hide.

This is practical in the IR spectrum but not the visible spectrum because humans would be very bothered by a bunch of white LEDs lighting up the scene from a camera. However, since they can't see infrared, they don't might a whole ton of IR illumination coming down.

In practice, some of the light from the IR leds is visible as a dull red glow, which shows up pink when photographed due to the way digital cameras process their images.

Absolutely, depending on what you mean by eyes that is. If you mean the camera that takes in information, then no, but if you mean the area on their face where eyes would be, then yes. In the movie Chappie, the robotic police force use a display screen where their eyes would be to display information for the humans running them;

But the screen can really display anything, including eyes that, because it's a screen, glow. The reason why the robot would display eyes is another question, but answers include: human-robot relations, appearing less threatening and artificial intelligence.

If by "eyes" you mean the cameras or whatever sensors it uses to perceive the world visually, no. This is usually done in media for no better reason than because it looks cool.

In real life, many robots that are built for human entertainment have what seems like glowing eyes. But they don't see with those eyes. Take AIBO for an example:

At first glance it seems like the head lights are its eyes. Look again.

A robot could have glowing "eyes" so that a human talking to it will maintain proper eye contact, for whatever reasons the bot's designer had in mind - in AIBO's case it can use its LED's to communicate. Or they may work as flashlights (for humans, not for the robot) so that one can see which direction the bot is looking at even if they are behind the bot. Or they could serve as an indicator that the bot has your attention, or if it is turned on (i.e.: if the eyes are off, the robot is off or in stand-by).

Last but not least - the robot may be secretly communicating using a protocol similar to morse code*. They are blinking their eyes too fast for us to see. Think about it, a fluorescent light "blinks" 60 times per second and we can't see it. The robots have been telling jokes about us humans all this time right under our noses, next thing they are going to break the first law and kill us all.

*Yeah, they could use radio to discuss their plans via bluetooth or wifi, but they knew we'd be snooping that kind of communication.

In the novel Robopocalypse (which is really just World War Z with robots instead of zombies) robots have "intention lights" which glow specific colours to communicate to people whether they understand the instruction, operational status, etc. These are simple non-humanoid bots with basic personalities, but as I recall the more advanced bots with "faces" retained this legacy feature.

Your bots might have something similar, coded lights that communicate the equivalent of facial expression or status to people around them.

My webcam has a light, as to most video camers to indicate they are recording (or where to look while being recorded).

So the reason is a social convention or legal requirement: a tally lamp to indicate that a machine is in fact looking at you, and an indication of how to make “eye contact” because people are uneasy without that.

The eyeshine of certain vertebrates was an evolutionary side-effect, when these animals evolved in an environment where artificial lighting was not available at night. Therefore eyeshine cannot be compared to the glowing eyes of robots. (This side-effect must have saved countless humans who were fortunate enough to gather around a fire).

However, there's a perfectly valid reason to have glowing eyes for humanoid robots, or robots who interact vocally with humans and have a face. Humans would be far more comfortable talking to a "face" with eyes rather than a globe of light.

Is there any plausible explanation as to why a future robot's eyes might glow?

Reason #1: Because it's scary.

If the robots are built with the intention of being intimidating or frightening, then they will be designed with scary-looking features. Glowing red eyes are great for that, especially if combined with other evil-looking facial attributes. They'll probably also have bulges to make them look muscular, random unnecessary spiky bits, lots of dark coloured brushed metal, and other industrial design cues.

Welcome to the future! Muahahaha!

Well, "future robots" probably not, but when we revert to steam punk, it's worth noting that a number of historical video camera tubes actually featured a glowing cathode for the emittance of the scanning electron beam.

Of course, it would be highly unlikely to see that glow frontally (rather than at best from the side of the tube) since the optical path to the charge plate (sort of like the retina) would not intersect with the cathode for the scanner.

Where the "retina" itself is a cathode (like with photomultiplier tubes), it's much more likely to require cooling than heating in order to reduce electron emission not prompted by light.