There are several policies documented in GitHub Issue Management that the on-call platform reviewer should enforce. The on-call reviewer should run through this checklist at least once per day.
Search for issues without an assigned team and assign a team. When in doubt, seek advice on slack.
Search for pull requests that need review and (probably) assign a feature reviewer. This is intended to make sure that requests from outside developers receive timely attention. For a pull request by a core Drake Developer, leaving it unassigned may be acceptable when it is clearly an early work-in-progress – but if it is unassigned for several days, you should probably encourage the developer to label it “do not review” for clarity.
For PRs assigned to you that have passed all commit checks (other than needing a squash), merge the PR to master on behalf of the author – unless it is labeled “status: do not merge”.