Is there any conceivable way that a planet with two moons--one being visibly red to the naked eye due to its mineral composition--could align in an eclipse in such a way so that the whole effect looks like a glaring eye in the sky, with the smaller red planet as the "eye's" pupil?
So you have 2 moons, a red one and white one.
The red one has to have a smaller orbital radius than the white one, so that it lines up "in front" when viewed from the planet's surface. Additionally, it has to be smaller than the white one so the white one isn't concealed behind it when it transits. For a stable system the two moons' orbital radii can't be too similar or they'll pass too close to each other and attract/crash into each other eventually, so the outer white moon has to be significantly further out, and so (significantly)^2 larger than the inner red moon so that it still has a broader appearance in the sky.
If you have a small red moon on the inside and a big white moon on the outside, you'll get your red glaring eye, but it'll look like a googly eye (just white and pupil, no iris).
If you want an iris you need another disc. You could have the sun as the white of the eye, with a white iris and red pupil, but in an eclipse both moons will just look black, so that's not a great idea. For a full eye-iris-pupil system you could have 3 moons (the white, the red, and a third black one, even smaller and even closer than the other two) which line up to give a glaring red eye.
Alternatively you could have just the white and red moons, but they're really big and low-density, while your planet is extremely small and high density, such that when the sun is behind the planet and you look up at midnight, you see your planet's shadow as a round dark pupil on the red moon in front of the white moon, an angry eye glaring down.
I'll take an idea from Henry Taylors' answer. I think that the converging moon would appear black as it transited the face of the larger, outer moon. But lets assume that the small moon is tidal locked and has a volcanic core that has emitted a permanent a lake of magma near the center of the moon. This would be invisible or nearly so when illuminated by sunlight.
So when the two moons coincide near full, the black moon appears in the 'white' background. Then when the lake appears you see the glow of the magma pool and you get concentric white - black - red.
First set the size of the first moon by deciding its radius in degrees. That is, the angle between the lines from to point of observation to center of moon and the edges of the moon. Next set the size of the second moon by getting its radius in degrees, which is simply what you decided for the first moon plus how big you want the iris to be. It becomes quickly obvious from simple geometry that the mechanics are easier if the angles are smaller. Since no lower limit on the visible size was specified, it is possible to have the effect.
Apart from the visible size another important variable is how wide you want the area the effect is visible to be as that controls how far the moons need to be. Or the ratio of the distances between the three objects, really. Another consideration is the need for both moons to be full at the same time, which sets a minimum separation for the objects. Although the minimum distances set by orbital stability and tidal forces might be larger.
Honestly there are so many variables there is no point doing the math manually. There are probably some astronomy programs tha can simulate the model, show what it would look like, and allow you to tinker with the values until the desired result is achieved.
You'd probably want the larger moon to be relatively dim for the red to be more spectacular.
The main issue is illumination, as it has already been pointed, but not fully accounted for.
You can have the two moons in different planes, and have them coincide only in the nodes line. This actually happens at Earth for Sun and Moon: they do not eclipse every 14 days, since they are not in teh same plane, but only when they both are at the nodes line at the same time.
In the case at hand, you can have three different planes for the two moons and the sun. The one for the sun is very important because it allow the moons to be (almost) fully illuminated from behind the planet without casting any shadow, not the planet on them nor the smaller moon on the bigger.
So for your case, you have a line with the two moons and the planet, for the eye effect being created, and the sun out of that line, illuminating the three bodies. This also implies the effect can happen only over the dark side of the planet (that is, at night). Unless there are more than one sun that creates an additional daytime.
I recommend you to test the case with Universe Sandbox ² or any other simulator.
Yes, you can
I think the color pallet might be inverted, but if the smaller moon somehow glowed then a four-way convergence (sun, big-moon, small-moon, planet) might appear as a giant black eye with the sun's corona glowing around the edge and a smoldering red pupil in the middle.
The trick would be to explain the small moon's glow. I defer to the gravity gurus that frequent this site, but if the small moon could have a molten core and a thin mantle, perhaps the alignments of gravities during the convergence could set off huge caldera volcanos on the planet facing side of the moon.
If the moon had no atmosphere, I can further imagine that the dust clouds from the volcanos might escape its gravity well and disperse quickly, leaving the fires below clearly visible from the planet's point of view. That part probably doesn't work, since during convergence, the big moon would be blocking any solar winds vectoring out from the sun, so maybe the aligned gravities would pull the dust clouds back down into the lava instead. Again, a task for the gravity experts to figure out.