Connors - who has a chequered off-field record - mutually agreed with the Tigers to end his time at the AFL club after failing to front up for the practice session on Tuesday.
He was on prescribed sleeping tablets.
A remorseful Martin, one of Richmond's most high-profile players, admitted to drinking alcohol on Monday and taking unprescribed sleeping tablets.
Richmond's general manager of football Craig Cameron said Connors had decided to walk away from Richmond and focus on the next phase of his life.
"We have set a standard as a football club and there is a clear expectation that our players meet that standard," said Cameron.
"Adhering to training requirements is not negotiable.
"Daniel has failed to meet club expectations on a number of occasions and this latest incident can't be considered in isolation.
"Both the Richmond Football Club and the AFL Players Association have fully supported Daniel as he has worked through some well-documented issues, and we will continue to work with him while he focuses on his wellbeing."
Connors played 29 senior matches for Richmond, but only two in 2012.
Martin, 21, has played 56 matches since being selected by the Tigers with pick No.3 in the 2009 national draft.
He will be suspended for the Tigers' must-win games against Melbourne and Gold Coast.
"Dustin understands that missing training is simply unacceptable," said Cameron.
"We want to send a strong message that we don't accept this sort of behaviour and Dustin is now fully aware of this.
"More broadly, our medical staff will continue to educate our entire playing group and emphasise that prescribed medication needs to be taken as directed, and the significant dangers associated with not doing so."
The situation could hardly have come at a worse time for 11th-placed Richmond, who may need to win as many as seven of their last nine matches to claim a finals berth for the first time in 11 years.
Midfielder Daniel Jackson will miss the next two weeks after being suspended for striking Adelaide's Chris Knights.
Small forward Jake King (knee) and key defender Dylan Grimes (hamstring) both face significant spells on the sidelines after being injured in Saturday's loss to the Crows.
And it was revealed on Tuesday that ruck-forward Ty Vickery will undergo a shoulder reconstruction next week, ruling him out for the remainder of the 2012 campaign.