The first time I remember hearing the term "innings eater" was when the Phillies traded for Joe Blanton at the 2008 trade deadline. In that particular case, the term was employed as a compliment; the version of Joe Blanton the Phillies acquired was a league-average or slightly better pitcher and threw at least 190 innings in each of the three years prior to arriving in Philadelphia. Just about any team would be happy to have Joe Blanton eat innings for them as a number four or five starter.
Eating innings means something different for a good team--a team contending for the World Series, no less--than it does for a bad team.