Juventus winger Chiesa limped off after coming on as a substitute as Juventus beat Inter Milan in a Serie A.
Examinations on his knee showed no injury, but the club said he needs to rest.
Inter left back Dimarco went off earlier the same game.
West Ham defender Emerson Palmieri has been called up for Dimarco but no replacement was named for Chiesa.
Chiesa was a starter on the team that beat England on penalties in the last final in 2021.
Ivan Provedel of Lazio, one of four goalkeepers called up by coach Roberto Mancini, also withdrew and was replaced by Cremonese's Marco Carnesecchi.
Italy begin the defence of their title with a rematch of the Euro 2020 final against England in Naples on Thursday, before travelling to Malta on Sunday.