In one corner of my yard I have two largish trees that are about 5 feet apart. One is a box elder, the other is a norway maple. They both have trunks about 12-14 inches in diameter. The box elder is a little bigger and taller. It seems like the canopies of these trees are interfering with each other. Are they too close? Should I cut one down?
Definitely too close together - the crown of a mature Norway Maple (assuming you mean Acer platanoides) is more than 25 feet and the crown of a Box Elder is (if left unpruned) 25 to 40 feet, so they will crowd each other badly.
You can decide to take one out, or just keep the Box Elder constantly pruned, more like a shrub than a tree, though that won't be easy...probably best to remove one.
I am probably a little late with my answer, but I want to bring to your attention another idea:
If both trees appear to be healthy, it is a shame to kill one of them. Instead, you can coppice the smaller one - norway maple. This means you cut it back to ground level (or a couple of cm/inch above) in dormant season (winter), and then wait for regrowing. Be patient, the regrowth may happen later in the spring, in May or June. This is possible with norway maples - I did it once! The maple will become a multistem shrub in a couple of years, without your intervention at all. In seven to twelve years you can repeat the procedure. That way, the crowns of box elder and norway maple won't interfere at all. Underground, of course, they will compete, but I think that competition is already sorted out, and they both have they own territories, so there is no really big need to worry about it. Good luck!