This is probably more of a tSQLt question, so it would be better to ask at the tSQLt forum
. It does look like you can do it, though, by installing SQL Test on the new server (if you are going to a new server), add the database to SQL Test, and copy the schema and data to the new database.
The tests you write are in a schema identified by the class name you give to SQL Test and the list of tests and results are stored in the tables and views inside the tSQLt schema for the database.
I'd check with the tSQLt people before doing anything rash.