LaCosse, 27, was entering his second year with the Patriots and sixth in the NFL.
His wife, Jessica, is pregnant and due during the season, a source said.
In addition to LaCosse, other top Patriots players to opt out of the 2020 season are starting linebacker Dont'a Hightower, starting safety Patrick Chung, starting right tackle Marcus Cannon and core special-teamer Brandon Bolden.
Fullback Danny Vitale, center/guard Najee Toran and wide receiver Marqise Lee, all of whom were uncertain to be on the initial 53-man roster, also opted out.
As for LaCosse, he had signed a two-year contract worth up to $3.8 million with the Patriots in March 2019. The second year of his contract, in which he was scheduled to earn a base salary of $1 million, will toll to the 2021 season.
Also, running back Sony Michel (foot), wide receiver Mohamed Sanu (ankle) and core special teamer Brandon King (quad) are opening Patriots training camp on the active/physically unable to perform list, a source told ESPN's Field Yates. The players can come off the list at any time.
LaCosse played in 11 games last season in New England, totaling 13 receptions for 131 yards and one touchdown. He was projected to be on the roster again this year, complementing rookie tight ends Devin Asiasi and Dalton Keene, both of whom the Patriots traded up to select in the third round of the NFL draft.
The position was a top priority for the team in 2020. Patriots tight ends ranked last in the NFL in receptions and targets, according to ESPN Stats & Information, and New England was tied with the Chicago Bears for the fewest receiving touchdowns from tight ends (two).
On Friday, when the total of Patriots opt-out players had reached six, Patriots coach Bill Belichick said, "I respect all of them, and I respect all the players on our team. We all have to make decisions. I talked to those guys, and they explained their situation and they had to make the decision that was best for them, and I totally respect and support it 100%."
Those deemed to be at high risk for the coronavirus receive $350,000 and an accrued NFL season if they opt out of the season. Players who aren't deemed to be at risk but don't feel comfortable playing can receive $150,000 if they opt out.