A .TDC file needs to be on the Tableau Server as well as on the developer's computer. The TDC file contains the necessary settings for the specific connection only and will only be invoked for that type of data source -- not the specific data source but the type e.g. MS SQL Server or GoogleAnalytics or Oracle or etc. The location on the Tableau Server varies with the major version, however, for Tableau Desktop you save the TDC file in the DATASOURCES folder of your Tableau Repository location. For example, here's where I have mine:
As you can see, this TDC is only for MS SQL Server connections.
Tableau doesn't support TDC files in the aspect like they do with their primary products. Thus you could ask for help and suggestions but they are not obligated to find you a solution. You will likely need to do experimentation yourself to figure out what works.
Toby, nice response. I just wanted to clarify that when using and fooling with .tdc files, you are pretty much on your own. Tableau provides the function to be able to control certain aspects of the data source connection, but you would not or should not expect to call into technical support and get help fooling with the tdc settings. That being said, there are times that a support engineer may ask you to open the tdc file and do something as part of a tech support case. Just be aware this is outside the normal Support position on tdc files.