No, you don't push a local branch to a central repo, unless it is actually needed as branch for someone else to see (i.e. for shared work). And if you did for whatever reason, you delete the branch once it was fully merged.
Git branches serve as heads and you use them for that. After merge the history will plainly show what branch was there, so keeping it would be redundant. And if you want to continue the branch, it can simply be re-created at the last commit. IOW deleting/not pushing it loses no info whatsoever.