Yes, you can wash felt. If you don't know whether it's felted wool or the kind of felt they sell at JAF made from recycled plastic, I'd stick with cold water, low heat & limit harsh cleansers. Either one can yellow with age. Felted wool is more likely to return to something closer to white, but both usually can improve. If you know or strongly suspect it's felted wool (usually cannot see through it very well when held up to light), it might be worthwhile to try soaking it with white vinegar. Add 1/4 C of vinegar to your washing machine & set on Soak cycle, Cold Water. When finished, wash 2x with regular detergent to get the vinegar smell out, then dry on low heat.
If you know/suspect it's the plastic felt (thin, easy to see through when held up to light), vinegar isn't likely to help. You're better off with Hydrogen Peroxide. H2O2 is more corrosive than vinegar, so I'd first try just 1/8C of it, let it Soak, then wash in Cold Water, but maybe add some OxyClean if you have some. If that doesn't work, you could also try soaking overnight in a laundry sink with a scoop of Biz.
You could also try H2O2 with felted wool, but just keep a careful eye on it & know that it will damage the fabric, even if the damage isn't visible right away. The longer it sits in that solution, the weaker the wool fibers will become. I definitely would not let it sit in that solution for longer than 20 minutes for any reason unless you're okay with it potentially burning holes in your felt. Not saying that will definitely happen, but it's a risk. Only you know how bad the yellowing is & whether it's worth it. Whatever you try, DO NOT try bleach on felted wool. It will make the yellowing worse. Some people also claim Baking Soda helps reduce the yellowing, but I don't have any idea how much you'd need. I'd probably try 1/4 C, but I don't think more would harm anything ... just might take an extra trip through the wash to get it all out.
For future reference: plastic felt turns yellow when exposed to sunlight & oils from our hands so wearing gloves & covering with a sheet when not in use will help. Felted wool turns yellow when exposed to sunlight & moisture -- moisture especially. No easy fix for that & felted wool is quite expensive. If your studio's in the basement, you could get a portable dehumidifier ... or just swap out design wall for cheaper fabric.
Good luck! Hope you can get it back to an acceptable color.