I thought nothing of doing it, but also feel it is a good question to pose to the posse' here. Wood that is seasoned multiple years, but was stacked 2 weeks ago uncovered near the house. Rained for a couple days so the stuff was soaked. I did not flinch and chucked it in the stove. Not an issue??
When I stack on the front porch, rain and snow will get to it. I'll bring it into the stove room and let it sit for a while, then stick it in the stove. Gooder to go. It's never soaked though. I burned fresh c/s wood for 2 years before I could get even slightly ahead, so It can be done.....just not well. What you're describing is different than that.
Seasoned multiple years? Split or unsplit? In this part of the world, we have several species that will just rot if the rounds are not split, and in most cases have to stacked in a well ventilated area, under cover from wind driven rain if you want the the splits to dry.
I'd get it top covered asap but it should dry off quickly; usually within 24-48 hours. No, it won't hurt the stove but if you have a cat stove, then you may need to leave the bypass open a bit longer but some say don't use it in a cat stove until that surface moisture is gone.
Wet wood for me goes under cover in a wheel barrow before coming inside, then lives by the hot stove for a while, and then into the stove. Surface moisture dries quickly as others have said. Especially when next to a warm stove. I like the moisture off the wood sometimes. Helps relieve the dry air in the winter.
Most of my wood has been scrounged and I don't have a great way to get rid of stuff that I wish I hadn't brought home. I try to make sure this stuff gets under a roof because it can be a PITA when it gets rained on. No issues with the solid stuff.
You burn what you got. Burn wet wood on high longer to evaporate the moisture out before adjusting the stove for the long burn. If you burn wet wood on cooler exhaust temps, just realize you are making creosote & need to clean & inspect the chimney often. Shoulder season burning at low stove & exhaust temps , burn good dry wood & you'll get less (or no )creosote build up.