Podolski joined up with the squad on Monday but further tests showed the Galatasaray winger had inflammation of the ankle that ruled him out.
"He has since left the team and will receive further treatment at his club," the statement said.
Fellow midfielder Karim Bellarabi is also being treated for a shoulder injury but could still be fit in time for Thursday's Group D game in Dublin that could secure Germany a spot in next year's tournament. Germany then host Georgia in Leipzig on Sunday.
The Germans have enough quality with Marco Reus, Mario Goetze, Andre Schuerrle and several others able to play in Podolski's position, but little time to prepare for the game on Thursday with the team having only one full training session with all the players present.
"We are a bit used to that," said playmaker Ilkay Guendogan. "You still need a few days for everyone to reach the same level, for everyone to arrive. Someone played on Friday, others played on Sunday.
"It is a bit difficult to coordinate, but what matters is that everyone is at 100 percent when the game starts," added the Dortmund midfielder.
Germany top their group on 19 points, two ahead of second-placed Poland and four ahead of Ireland, who are third on 15. A win or a draw would secure them a top two finish and qualification for France.
"I do not think that we will go out to play for just one point," Guendogan said. "With our demands and our quality we want to win the game and do it as we did it in our (3-2) win against Scotland."
(Reporting by Karolos Grohmann, editing by Pritha Sarkar and Toby Davis)
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