You can shoot with both eyes open when using any type of sight, but it's easier with some than others. The glowing reticle in reflex and holographic sights makes them easier and faster to acquire. Some people shoot the same way with telescopes (search Binden Aiming Concept, A.K.A. "BAC"). Trijicon and others make telescopic sights with glowing reticles for just that purpose (generally, they're 4x or less).
The NRA training classes I took back in November had us sighting through iron sights with both eyes open, to improve acuity.
G. David Tubb (a prominent Highpower champion) wrote a book in which he describes using a piece of cellophane tape on his glasses over the non-aiming eye when shooting with iron sights. It just made it easier for him, and since I'm cross-dominant I do the same thing sometimes. It closes your aiming pupil more than with one eye closed (your pupils tend to work in unison off of a sum-light value) and that improves both yourt depth-of-field and your acuity.
With a reflex sight, there's no need to close an eye or to use the tape for any reason, because everything is on the same focal plane and the dot is so easy to acquire. You can in fact even use a reflex sight with the front lens cover closed (search Occluded Eye Gunsight [OEG]). Aimpoint actually recommends doing the latter as a way of familiarizing the shooter with the two-eyes sighting method and I concur.