If your drain analogy holds beyond the point of you losing the sightof probe, the matter that goes into drain does not cease to exists, even if for all your practical purposes it does. On the contrary, you can be sure it does reappear somewhere.

Also, it's pretty much certain that wherever there appears infinity in physical model you can be sure it's a limitation of the model, not actual physics of the object. We do not observe infiities, even though pretty much every of our models has some.

Those would be two basic hunches against stuffing matter into infinity without consequences. Some kind of exit seems natural, the question is how destructive it would be.