As a result, Jones was fined an undisclosed amount and excused from all team activities, the team announced through a press release Sunday.
Shortly after Sam was drafted with the 249th overall pick in the seventh round and became the first openly gay player in the NFL, Jones tweeted "omg" after television cameras showed Sam kissing his boyfriend.
According to The Miami Herald, when Jones was asked on his social media channel if he was referring to the embrace, he responded: "Horrible." The tweets have since been deleted.
The Dolphins said Jones cannot return to the team until he "undergoes and completes educational training."
"(The tweets) were inappropriate and unacceptable, and we regret the negative impact these comments had on such an important weekend for the NFL," coach Joe Philbin said in the release. "We will continue to emphasize and educate our players that these statements will not be tolerated."
Jones subsequently expressed regret for his actions.
"I want to apologize to Michael Sam for the inappropriate comments that I made last night on social media," Jones said in the statement. "I take full responsibility for them and I regret that these tweets took away from his draft moment.
"I remember last year when I was drafted in the seventh round and all of the emotions and happiness I felt when I received the call that gave me an opportunity to play for an NFL team and I wish him all the best in his NFL career."