We will be assuming lead development responsibilities for a bioconductor package that has already been accepted. What permissions are needed from the original author to continue development on this package? The package is currently in GitHub.

I have already read through the Contributions information, and it seems that packages can be updated just by pushing to the master branch with a version change. So, is it sufficient for us to have admin rights to the repository? Any other steps that need to be taken?

This is a question for the bioc-devel mailing list. Developers may maintain their package it git / github, but are ultimately responsible for make the code available via our cannonical svn repository. You'll need svn credentials, so you or (preferably) the original maintainer should send email to packages at bioconductor.org with your name, email address, and package name.