There is a feature (associated revisions) in Redmine to link commits with tasks with specific keywords typed in comment on commit:
Default keywords ar
Log in as an admin and go to Settings -> Repositories -> Referencing and fixing issues in commit messages:
Make sure that:
The following examples will close an issue for the above settings:
fixes #1234
closes #1234
To have the issue get closed, redmine will have to read the latest commit messages. Usually redmine does this when clicking on the Repository tab. You may also configure a rake task which does this regularly.