Apply too many schools? Prof tired of sending out recommendation letter?
My experience is that once the letter is written, it takes about 5-10 minutes to submit it to each institution.  Usually there is a short form that has to be filled out with basic information (name, contact information, etc) and sometimes also a list of questions asking the professor to rate the candidate in a variety of ways.

So you probably don't want to apply to a huge number of schools (for many reasons besides just that).  It's hard to be precise, but I feel like less than 10 is no problem, and 20 is probably too many.