This scenario comes up now and then and we've updated our OnlineHelp to try to make moving Tableau Server to a different drive more clear:
The part that catches most folks off guard would be step #6
6. Uninstall Tableau Server using Control Panel and Add/Remove programs. You should also rename the
\Program Files\Tableau\Tableau Server\<version>directories so your new installation does not find them. By default the
\ProgramDatadirectory is hidden, so you may need to configure Windows to display hidden files and directories to find it. Later, once you have installed Tableau Server to the new location and verify that everything is working, you can delete the directories.
In regards to space available it might become an issue when it comes to processing a backup file. You can use the "-t" option to specificy a different location (one that has more room) since all the extracts are copied to a temporary location before they get compressed and become part of the *.tsbak file: