"The Rag Thing" by Donald A. Wollheim

First appeared in F&SF.  Reprinted (once in a major genre retrospective) by Groff Conklin, Terry Carr, Martin H. Greenberg, Isaac Asimov, Al Sarrantonio, Robert Weinberg, and Stefan R. Dziemianowicz.

Landlady leaves her "cleaning rag" out (presumably, it is not a rag) last November.  The landlady turns off the heat too soon in March, so that the rag seeks out warmth.

Once again, we have the misunderstood single man in the apartment.  This time he set his apartment ablaze, but why?  They chalk it up to careless smoking in bed.  A new woman buys the boarding house....

Perhaps this explains why landladies shouldn't turn the heat off too soon.