Instead, lefty Henry Owens, Joe Kelly and Wade Miley will start in New York.
Rodriguez will rejoin the rotation at some point next week, when the Red Sox return for a nine-game homestand.
Between Triple-A and the Majors, Rodriguez is at 140 2/3 innings this season.
"We're trying to manage his time and his workload, and this is a way to give him a couple extra days. It makes sense for us," said Red Sox interim manager Torey Lovullo.
Owens is at 146 1/3 innings this season.
"I think both of these guys are getting ready to pitch into their sixth month of baseball, and they've never done that before," said Lovullo. "We're going to add everything up and make sure we don't put either of those guys in harm's way. Henry's name has come up a little bit, not as much as Eduardo. We'll let them dictate how they're doing on the mound and move forward with some of our thoughts."