The right-hander was placed on the 10-day injured list Monday with right middle finger inflammation and is expected to miss at least a month.
AL Cy Young Award winner Blake Snell has been out since July 22 with bone chips in his left elbow that required surgery.
Tyler Glasnow started the season 6-1 before being sidelined May 10 due to a right forearm strain.
The Rays have not ruled out all three returning in September.
Tampa Bay manager Kevin Cash said the team "is processing" how to handle the injuries.
"Certainly, when we've got three guys that we count on for 180-plus innings and they're all sitting and watching games, not ideal," Cash said. "But bet on these guys to pick each other up and find a way to figure it out."
Tampa Bay currently has Charlie Morton and two-way rookie Brendan McKay in its rotation. Ryan Yarbrough and Jalen Beeks have thrown multiple innings out of the bullpen when the team used a reliever as an opener.
Chirinos won his ninth game in a victory Sunday over Miami but was forced out following five innings after first experiencing pain in the finger while throwing a slider during the fourth.
"Of course it's sad, no one ever wants to get on the IL," Chirinos said through a translator. "But it's part of the game.
Chirinos will be shut down from throwing for two weeks.
Glasnow played catch for the first time in six weeks at 45 and 60 feet on level ground. He hopes to work off a mound in two weeks.
"It was only day one but it feels really, really good timing wise," Glasnow said. "It's recovering really well. I'm optimistic about it."
Snell said Monday that he had five little bone chips and one that "was pretty big" removed, and that doctors are now waiting for swelling to diminish before establishing a time frame for a throwing program.
"My range of motion is better than it was," Snell said. "Once the swelling goes down, then everything ramps up."
Tampa Bay started the day with a half-game lead over Oakland for the second AL wild card. The Rays won their sixth straight game Sunday, and set a team record by scoring six or more runs in seven consecutive games.
"We're nasty," Snell said. "We're swinging really well right now so it doesn't really matter who's on the mound. We'll be fine."