Redmine

You can configure which roles can change issues to select status. Administrators can configure this in Administration -> Workflows. Here, you can deny some roles to set the issues to a closed status.

The ability to delete issues is also defined by roles, this time in Administration -> Roles and permissions. Here, you can can define a generic "Delete issues" permission (or rather: remove this permission). Generally, I'd recommend to remove this permission for almost all users. Instead of deleting issues, it is generally a better idea to just close them. That way, their history is retained and they can be re-opened or checked later if required. Note that Administrators always have all permissions and thus wil always be allowed to delete issues.