LaCosse and his wife are reportedly expecting their first child in November, per Michael Giardi of NFL Network.
He becomes the eighth Patriots player to opt out amid the coronavirus pandemic, by far the most of any team in the NFL.
New England will also be without safety Patrick Chung, linebacker Dont'a Hightower, right tackle Marcus Cannon, fullback Dan Vitale, running back Brandon Bolden, offensive lineman Najee Toran and receiver Marqise Lee this season.
"I respect all of them," head coach Bill Belichick said of the players who opted out, via Kevin Patra of NFL.com. "I respect all the players on our team. We all have to make decisions. And I talked to those guys, and they explained their situations. They had to make the decision that was best for them, and I totally respect and support it 100 percent."
LaCosse only caught 13 passes for 131 yards last season, but he started eight games and played 38 percent of the team's offensive snaps in 2019, per Pro Football Reference.
The 27-year-old totaled 24 catches for 250 yards in 15 games with the Denver Broncos in 2018.
With Ben Watson retired, Ryan Izzo is now the only returning tight end from last year's Patriots squad.
The Patriots selected Devin Asiasi and Dalton Keane in the third-round of the 2020 draft, and both rookies could now be looking at major roles.