Jones took to Twitter soon after Sam was drafted by the St. Louis Rams May 10 and tweeted "OMG" and "horrible." He later deleted the tweets.
The Dolphins fined Jones an undisclosed amount, required him to undergo educational training and excused him from the team following the tweets on Sam, who became the first openly gay athlete drafted by an NFL team.
Jones apologized to Sam and Dolphins owner Stephen Ross shortly after posting the tweets.
Jones was drafted by the Dolphins in the seventh round in 2013. He was one of Miami's top special-teams players and finished with 10 tackles last season.
Information from ESPN.com Dolphins reporter James Walker was used in this report.