There's a ton of disagreement on this... but, I will say, I've never seen mold or mildew on my firewood (unless I've left it laying in the woodlot too long).
Sometimes, just before snow and ice hits, I'll lay some old boards or something on top... just so it's easy to throw the snow off.
Other than that... nope, don't cover, never have. That's not to say I haven't been caught needing to move some wood in the house during a 5-day rainy spell (usually in spring), but it's more of a nuisance than anything... it drys in about a day once it's in the house. With that said, I can put several cord in my basement just before heating season... if I couldn't do that I'd likely cover the stacks sometime in late fall/early winter (but I'd only cover the stacks I planned to use that year). I stack single rows, in near full sun, in an open area with plenty of breeze/wind... even after a week of rain, the splits are dry except the top couple layers or so (and they get the most sun when it comes back out, usually dry within a day). Honestly, until I joined this site I'd never heard of covering un-seasoned firewood, or putting it in a shed... covers and sheds have always been for fully-seasoned firewood in the world I live. If someone drove by and saw me covering "green" firewood they'd think I needed to spend a bit of time in a padded room ‼
Oh... and you're a whole lot less likely to have critters, varmints, vermin, snakes, insects and whatnot, take up residence in your firewood if it's stacked in well spaced single rows, with the sun beating down on it. In short... the wood stays nice and clean.
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