Someone recently told me that when you use iron sights, you're supposed to keep both eyes open. This seems odd to me as I was brought up to close one eye. The sights, at least the rear sight, tends to get a little blurry when I do this, and I have a hard time imagining it could ever be as accurate as one-eyed shooting. The only times I could see shooting both eyes open is in a reflex shot situation, like home defense or clay shooting where speed is paramount over accuracy. Or am I just wrong here...

I am right handed and my right eye is dominant so I have no trouble focusing on the front sight in normal shooting situations. PPC handgunning in the weak hand and weak eye stages I closed or at least squinted my right eye as I saw the sights come into my lower perifereal vision. In service rifle and pistol competition I always shot with both eyes open.

This is not an absolute definitive answer like you hoped for. My meaning in all that is that you should try all suggestions and then use the one, or an adaptation, that works best for you.

It is better to shoot with both eyes open...unless your dominant eye is not on you strong side.
I am right handed, my left eye is the dominant eye. I shoot handguns right handed, left eyed [both eyes open but left eye does the sighting]. I shoot rifles right handed and right eyed. I just close my left eye as I align the sights or look through the scope. You loose some depth perception and light gathering this way, but it is the way you have to shoot if you are cross eye dominant.
If you are this way do not worry...Chas. Askins was this way too.... He shot ok.

If you're like me, with a dominant, but weak, eye on the strong side, you can help the situation by wearing a translucent patch about the size of a dime over your weak eye on your glasses.

It does two things. It allows you to shoot with both eyes open without the weak eye fighting the strong eye for dominance, and it strengthens the focusing ability of the strong eye so that it will eventually be much stronger than the weak eye and you won't need the patch.

I am right handed and while my right eye is the dominant one, it isn't strong enough to keep the left from sometimes taking over.

I'm lucky in that I'm right handed and right eye dominant. I shoot pretty much everything (handguns, shotguns, and rifles) with both eyes open. This seems to help me immensely when using a scope, as my left eye never leaves the animals while the scope is moving into position in front of my right eye. Anyway, it seems to work, so I stick with it.

I too am right handed-right eye dominant. Everything is done with both eyes open.

I am right-handed and left-eye diminant. For some reason, despite hours of trying to train myself to do so, I have been unable to make myself close my left-eye. I sounds weird, but it is one of those querks. I have taken to shooting left-handed with everything, and I usually close my right eye. I realize this takes away from deapth perception and limits your vision, but it seems to be the only way for me to get a sharp, clear sight picture. When I try it with both eyes open and a handgun or shotgun, the sight picture become blurry, even though the target remains in focus. I tried picking objects on a wall and sighting them with both eyes open, then checking my sight picture using one eye, and found I was consistantly about and inch off from across the room. Which is why I said it would work for home defense or speed shooting, but would prefer one eyed shooting for precision. Strangely enough, I don't seem to have as much problems with a scope. I am wondering if it is because it is a one demensional sighting plane.
I intend to practice both eyes open for our annual clay shoot this Fouth of July. Every 4th, my grandpa, brother, great uncle, and I go through an entire wheel barrel full of shells shooting clay pigeons. It is basically 2 shotguns firing nearly constantly for the better part of an hour. By the time we get done, we are standing on a 6 foot circle of spent hulls stacked 3 inches deep. My favorite time of the year

2 up to but not including 4X glass, for all guns.