[somehow I never submitted this and Steven put it much more succinctly]
There are several ways to save album art in MusicBee. All of them require some degree of manual input from you. MusicBee always takes a hands-off approach until the user asks for more.
To verify whether artwork is saved in the file, go to the artwork tab of the tag editor. If there's no image, it's not saved. If there is an image and you hover over it, it will either say "embedded" or it will give you the name of the file where it's saved.
I think your problem will be solved by going to Tags(2) preferences and turning on the option to save artwork automatically. If that doesn't do it, then either I've misunderstood or there's something going on that's not with MusicBee.