I would like to know if you cover your boards with foil that are under the cakes of a tiered cake, or do you just use the cardboard as is?

I like to cover them with wax paper.

I won't use wax paper because it disintagrates from the moisture in the cakes.  I've had too many bits of paper served w/cake before I figured that one out ;(    all my boards are covered w/freezer paper and if it is used between tiers it is covered to & bottom.  I cut the paper about 2-3" larger than the round, cut slits all around & fold it under & tape, then cut another round exactly the size of the board & glue that on covering the pleated taped side.

I do not cover my boards... 

Do they not get soggy?

My experience is it depends on the cake.  They do get very greasy therefore are not as strong - tend to bend.   And there is definately a carboard taste :(

for internal boards placed between tiers -- not me nor any place i ever worked covered the boards -- but for bottom boards yes always cover -- florist foil is often used and it is not food safe -- so if i use that i use a plain cardboard under the cake and hot glue it to the foil covered board -- also wilton has some food safe paper that  is pretty -- silver and gold probably more

my first choice is waxed boards --

I like to use the corrugated plastic boards from Hobby Lobby between tiers. They're more expensive, but really nice. When I use regular cardboard boards, I put Press n Seal on the unwaxed side.

I use parchment paper and cut the circles I need. I put a little buttercream on each side to secure it and then put my cake board on top. It has never gotten too soggy and it comes off without taking the buttercream off underneath.