Other Strings Ringing Out While Tapping?

Pick ups are too hot, try lowering them a bit, most people don't even know you can do that, they'll just go out and buy new pick ups rather than trying different pick up heights,

I personally don't "tap", but it seems to me that you should use the rear of your palm of the right hand to lightly rest on your E,A and D strings to mute them. I'm pretty sure that's how the guys who use that technique do it.

(I googled a bit after I answered you and found that, yes, that's pretty much what you do. Try googling "guitar tapping technique" or "muting open strings while tapping" that should give you a lot of sites that will show you how to do it right.)