The team made the announcement about two-and-a-half hours before tipoff to the Eastern Conference finals opener.
Smart suffered the sprain in Game 7 against the Bucks on Sunday, and had been listed as questionable for the series opener in Miami.
Horford's absence, meanwhile, comes as a total surprise, with no indication of any issue with the veteran forward until the team announced that he had entered the protocols. Celtics coach Ime Udoka said he initially found out about Horford's status around 5 p.m. Tuesday. He wouldn't say if Horford tested positive.
"As always, we don't comment on the status of our guys," Udoka said. "He's feeling fine. We'll go from there. Wait to see results and tests and future tests."
Sources told ESPN's Adrian Wojnarowski that the Celtics are preparing to be without Horford through at least Game 2 on Thursday night.
"It's unsure," Udoka said before the game, when asked how long Horford might be out. "It's more of a medical question, honestly. But there are different protocols and tests that have to be passed, and we'll know going forward, but it's not a definite that he's out for two games."
For a player to return to game action after testing positive, there are two paths: Return two consecutive negative PCR tests at least 24 hours apart; return two consecutive PCR tests sampled at least 24 hours apart with cycle threshold (CT) values greater than 30 beginning no sooner than Day 4 and 5 after initially testing positive.
This marks the third time Horford has been placed in the league's health and safety protocols. He tested positive for COVID-19 during the preseason, and then was placed in the protocols again in December.
Spoelstra, on all COVID issues popping up around the league again:
"It's disappointing. It's like every single time we think, 'All right, this is kind of getting behind us,' it's not, and it's frustrating for all of us," said Heat head coach Erik Spoelstra on Wednesday. "You don't want to see it on either side. You want to play against their best and try to beat their best with our best. It's just not the world we're living in right now. It's really been three years of this. So I would expect the unexpected."
Smart, the NBA Defensive Player of the Year, suffered the injury in Boston's Game 7 victory over Milwaukee, when he took a tumble over Bucks superstar Giannis Antetokounmpo.
News of the absences of both Smart and Horford had an immediate sports betting impact. The Game 1 point spread moved from Heat -2.5 to Heat -4 at Caesars Sportsbook.
The Heat, meanwhile, again were without Kyle Lowry, who was ruled out Monday due to a hamstring strain.