Take it to a dry cleaner.Most of them, particularly if you are a regular customer, will not charge you fot it. They have to remove the Ateret to clean it; ask him to not reattach it until you have a chance to use a silver polish on it
If your’re in the Bklyn area, Mr. Mendelovich on Ft. Hamilton Pkwk is the best in the business. He removes the atoroh, gives you a new lining, ect. The cost is roughly $60
Whatever you do, DONT wash it in the machine. I did it to one, and the white pulled thru from under the black stripes so it looked like it was covered with lint (but was unremoveable)
In general, try not to wash a tallis to often. The natural color of wool, is yellow. The more you wash the fabric, the more you will fade the white dye!
Once you are removing the atarah already to clean the talis, have Mr. Mendelovich mount the atarah onto a seperate piece of silk and then either sew on snaps or velcro to the silk and to the talis, so that next time you need to clean the talis, you can easily remove the atarah.