We are looking at upgrading, but management does not want to buy another server.
We currently have TFS 2012 installed. Can I install TFS 2015 beside it or will that create a problem?
I would imagine I would at least need to configure a different port for TFS web.
No you cannot. The installer for 2015 will force you to remove the 2012 Application Tier.
Your SQL Server database server will be shareable as long as it's a version that supported by both 2012 and 2015 (SQL Server 2012 (minimum SP1)). And as long as both instances have a different database label configured.