The Board's Estimation/Time Tracking option is set to 'Remaining Estimate and Time Spent'

1) Create a subtask of a story in the active sprint
2) It's estimated time should be 10 hours
3) Log 5 hours
At this point the burndown should correctly log 5 hours spent and 5 remaining
4) Resolve the issue (leave 'Remaining Estimate' set to Adjust automatically and don't log any additional time)
The burndown chart will not remove the last 5 hours from 'Remaining Estimate' even though the JIRA issue correctly resets the remaining time to 0h

The workaround seems to be to either log work when closing an issue, or manually set the remaining time to 0 by logging work before closing the issue. Once the issue is closed, it's a bit more work to fix the burndown chart (assuming you know which issues were improperly closed)