I am working on a back-end, closed-source system for my company, the code for which is never likely to be open-sourced. Is there any reason for including a copyright notice and year at the top of each code file as you find in many open-source projects, or is this unnecessary from a legal standpoint?
Always, always, always include copyright notices in your code regardless of anyone's intent to distribute or not to distribute the source code.
Policy may change in the future. Who wants to manually add a few hundred copyright notices or try to come up with a regex or other process to do it correctly? Just make it part of your "new file" template and be done with it.
While the software might be an internal system, never underestimate the desire of management to sell anything they can to generate more revenue. C-levels network and talk. Once one of them figures out "we have an internal system for X" and "company Y will pay us millions of dollars for a system to do X" then guess what? Time to market and sell/license the system!
Rogue employee may steal source code. Sure they could remove the notices, but the fact that your copy has it intact is usable evidence against said rogue employee.
Copyright notices and source control metadata (added on date X) can provide valuable evidence to prove ownership.
Inadvertent disclosure. Perhaps a customer is on-site and sees some code or accidentally gains access. The copyright notice and lack of a license makes it abundantly clear that the customer is not allowed to take or use the code.
In the U.S., copyright is implicitly assigned the instant it is created:
When is my work protected?
Your work is under copyright protection the moment it is created and fixed in a tangible form that it is perceptible either directly or with the aid of a machine or device.
Source: U.S. Copyright Office
There is no legal requirement to have a copyright notice in the case described in the question. However, unless the copyright is registered (unlikely in this situation), arguments about ownership could devolve into an argument where two entity's words are pitted against each other at face value. By affixing copyright notices to a work you make it known both to potential copyright infringers as well as a court of law that you (or your employer in the case of a work for hire) assert ownership over the work.
Owning a copyright is different from proving ownership of a copyright. Given the low likelihood of registering a copyright in the scenario presented in this question, it is even more important to assert ownership any way possible.