I concur with
@bigealta that it can get dry uncovered (barring a time right before burning).
I concur that tarps suck (hence I built a shed, but if you can find used corrugated metal, and use a few 2x4s like
@peakbagger does (can you put a pic up here ? Just to show a cheap way of getting a solid roof?), screwing them in your stacked wood, holding up the metal off of the stacks, that's a lot cheaper).
I concur that the wood that is most sun and wind exposed (i.e. the top layer) will end up drier than the wood that (still gets wet when it rains, farther down below, but) does not see as much sun and wind.
My sole gripe with all of this is that it *will* be slower in drying (if compared with wood covered with a covering that does not touch the wood, see condensation and air flow there).
Why? Wood dries from the outside in. Getting it wet repeatedly necessarily slows the drying. No way around that.
Sure the sun won't hit the top layer of wood when covered. But uncovered the sun only hits that top layer of wood, and not much if any the lower layers.
However, it's not different from having a piece of wood heating up in the sun (from the outside in), and you repeatedly put icecubes against its outer surface. It will heat up slower.
So is it possible to dry the wood down to a level that is good for burning - evidently, as a lot of people do. Is it the best way? That depends on the effort you want to put in to make your stacks. No covering is lowest effort and cost. Shower curtains are cheap. Tarps are a bit more (and tend to leak). Used EPDM is nice. Sheds are expensive.
Finally, you mention trees nearby. I then strongly suggest you ensure that you get the leaves off of your wood in fall asap - leaves in nooks and crannies in your stacks will keep that wet way longer, and (leaves) will start to rot.
As I said in the beginning: wood drying approaches tend to get everyone involved with many different working (and nonworking) methods. Take stock of approaches. Find your own way, and get a moisture meter to check how your way is doing. If satisfactory, then you won't need that meter 3 years from now as you'll have found your way that works.