Which one of the following statements is correct:
- I deployed something to the test server.
- I deployed something on the test server.
I would say it depends where you are deploying. Deploying "to" the test server implies a remote operation - HTTP / ssh / ftp, amongst others. Deploying "on" the test server implies that you are at the console/GUI of the server, whether it be physically (touching the keyboard and mouse) or virtually (VNC or similar application that acts on the server itself).
Your examples require articles:
I deployed something to the test server.
Otherwise, these two are exactly equivalent.