Does one need to reduce the remaining hours to zero before moving the
task to "Done"? Or is it recommended to keep the hour estimate on each
task as it is, before moving the card over to the right?
Just personal opinion, I think one member should reduce the remaining hours to zero before moving the task to "Done" if the task is really done!
Please check Update remaining work:
Updating Remaining Work, preferably prior to the daily Scrum meeting, helps the team stay informed of the progress being made. It also ensures a smoother burndown chart.
Each team member can review the tasks they've worked on and estimate the work remaining. If they've discovered that it's taking longer than expected to complete, they should increase the remaining work for the task. Remaining Work should always reflect exactly how much work the team member estimates is remaining to complete the task.
So it's officially recommended to update the work remaining if the work is nearly done. Then the Team manager can manage members' work load accurately!