Device sleeves and cases (items that have a close fit):
Unless you're a pro-level wool owner, stick with spot cleaning. I don't suggest submerging or saturating a device sleeve in water unless you're prepared to block it as it may shrink a little as it dries. Lightly brush as much of the dirt, dust, etc. off as you can - you may even try a little vacuuming. If whatever has gotten on your felt is in liquid form, avoid rubbing it as it will make it fuzzy. Using cool or cold water, daub water onto the problem area only. There are wool specific detergents available if you're so inclined - very popular with knitters and can be found at your local knit shop or craft store.
On a personal note, I don't wash my sleeves. The natural lanolin content of wool keep them pretty protected and part of the reason I choose wool is because I like the way it lives alongside me. I've been using the same bag liner for the last 10 years and it still looks great. And I'm not an anomaly. There are lots of 6, 7, 8 (or more) year old byrd & belle sleeves walking around in the world looking good as new. The most I typically do is vacuum them.
Coasters, mats, etc:
If you don't mind if they shrink a tiny bit - you can soak these items in cold water. A bit of wool specific detergent might be a good idea - very popular with knitters and can be found at your local knit shop or craft store.
There's a few wool refresher sprays on the market but for a home remedy, try sprizing diluted lemon juice. Wool naturally smells a little when it's wet/damp so don't rush to this solution if you problem is just damp wool. Air it out instead.