Paul will travel with the Rockets to Oakland, California, for the game, coach Mike D'Antoni said, and will be re-evaluated after the team returns to Houston.
"It's obviously not something we wanted," D'Antoni said before Friday's practice. "I hate it for him above all. He's practically won us the past two games. But it's a great opportunity for other guys, and we have plenty to choose from. We'll be ready."
Said James Harden: "Obviously Chris has been playing extremely well. Hopefully [he will have] a speedy recovery, which I know will happen. There's an opportunity, like all year long, for guys to step up and take advantage of an unbelievable opportunity."
Asked if there was a chance Paul could return for a potential Game 7, D'Antoni said, "I don't know."
Paul exited Houston's 98-94 Game 5 win on Thursday in the final minute and received extensive postgame treatment, leaving the Toyota Center around midnight.
D'Antoni said Eric Gordon will start in Paul's place. Gordon averaged 21.5 points and 3.4 assists in 30 games as a fill-in starter this season.
"It will not change anything we do and how we do it," D'Antoni said, although he noted the rotation could expand to eight or nine players.
The Rockets have played only seven players in each of the past two games. D'Antoni said forward Luc Mbah a Moute would likely get minutes in Game 6; Mbah a Moute had fallen out of the rotation after struggling upon returning from a dislocated right shoulder for the second time this season.
"We're extremely confident, as usual, from top to bottom," Harden said. "This is nothing new to us. Whether it's myself or Chris or anybody else that has sat out a few games or whatnot, guys have stepped in and contributed very well."
A 13-year veteran who is one win from his first NBA Finals appearance, Paul reached for his hamstring after missing a runner in the lane with 51.7 seconds left in Game 5. He limped noticeably during the next two possessions and checked out of the game after briefly consulting with Rockets athletic trainer Keith Jones during a timeout with 22.4 seconds remaining.
Paul scored 18 of his 20 points in the second half to help Houston take a 3-2 series lead. He also had seven rebounds, six assists and three steals.
He has been dealing with soreness in his right foot since late in Game 2. He said the soreness diminished significantly between Games 3 and 4.
Paul has been sidelined by a hamstring injury on four occasions during his career, according to ESPN Stats & Information. His shortest absence due to a hamstring injury was two games during the 2015 West semifinals against the Rockets, when he played for the LA Clippers. Hamstring injuries sidelined Paul for five games in January 2012, seven games in December 2016 and January 2017, and three games in March this season.
The Rockets managed to go 15-6 this season in games Paul missed when they had MVP front-runner Harden and blossoming center Clint Capela available. Houston has outscored the Warriors by 11 points in 47 minutes during the West finals with Harden on the floor and Paul resting.
The Warriors opened as a 10.5-point favorite for Game 6 at several sportsbooks. After Paul was ruled out, the Warriors were listed as high as 12-point favorites over the Rockets.
Golden State remains a small favorite, minus-130, to win the series.
Information from ESPN's Tim MacMahon, Stefano Fusaro and David Purdum was used in this report.