If I have a couple American Silver Eagles with heavy toning (hey, they were cheap!) and I want to store them in the same plastic mint tube as untoned, great looking American Silver Eagles, will this increase the likelihood that the untoned coins will become toned?
no, toning is the process of the top layer of the coin being altered, environmental damage if you say, it will not go onto the next coin.
Toning is the formation of silver sulfide, it is strongly bound to the surface and will not spread. On the other side, verdigris WILL spread. Keep any coins with verdigris separate from clean coins.