The short answer is that no it should not because packets sent from your computer to a server on the same local network should never leave the local network assuming the network is sensibly configured.
However, the short answer makes another assumption about your network, which is that your local network is actually local. Many organisations now outsource services to the cloud and use various technologies to tunnel data packets across network boundaries (e.g. over the public Internet) while making it all look like the server has a local IP address.
Alternatively, the server could be in your network but at another location, such as a remote data centre or another office, which would similarly require packets to tunnel across the public network (Internet).
The upshot is that large file you're transferring to a file server on your network could actually be chewing up Internet bandwidth. The only way to know for sure is to ask someone who understands your network topology and knows for sure where the server is both physically and logically.