I connected bare #4 copper wire to two grounding rods. Is the #4 wire supposed to be insulated to prevent corrosion? Should I go back and put insulation around the bare wire?

 

No and no.

 

O.k., that was simple. How about covering the exposed bare #4 copper wire with dirt? Wife wants to hide the exposed copper, but I want to be able to check the connection once in a while. I'm just worried about corrosion.

 

It is fine to cover the wire and the top of the grounding rods with dirt. Some areas actually require the top of the grounding rod to be burried. As long as you got the clamp on good and tight, don't worry about corrosion.

 

What the others said. Don't worry at all about corrosion.
In fact IP is correct. Code says the end of the rod should be flush with or below grade.

 

If it is to be inspected, it would probably be a good idea to leave the dirt at the top of the rod "cupped out" so the inspector can see that you used the proper ground rod clamp and rod. But, as the others have said, the top of the rod should be at or below grade. Ground rods and property pins left too high sure are hard on mower crankshafts.

 

I hit one of these, with the lawnmower, that was raised up too high out in the yard, hidden by the tall grasss and it twisted the sheer key on the lawnmower shaft so the lawn mower couldn't restart until I replaced the sheer key. That got a few (&(*&*$##!!! words out of me.