Break the link between a Database User and SQL Server Login
You can accomplish this in a roundabout way be creating a temporary login, remapping the user to the temporary login, and then dropping the temporary login.  For a SQL login:

USE [master]
CREATE LOGIN [temp_user] WITH PASSWORD=N'asdf' MUST_CHANGE, 
    DEFAULT_DATABASE=[master], CHECK_EXPIRATION=ON, CHECK_POLICY=ON
ALTER LOGIN [temp_user] DISABLE

USE [test_database]
ALTER USER test_user WITH LOGIN = [temp_user]
DROP LOGIN [temp_user]


For a Windows authenticated login/user, you would need to create a temporary Windows account, and then delete it, so it's not a 100% T-SQL solution:

-- Create a Windows account with a name of Temp_User
USE [master]
CREATE LOGIN [COMPUTERNAME\Temp_User] FROM WINDOWS WITH DEFAULT_DATABASE=[master]
ALTER LOGIN [COMPUTERNAME\Temp_User] DISABLE

USE [test]
ALTER USER [COMPUTERNAME\Existing_User] WITH LOGIN = [COMPUTERNAME\Temp_User]
--Delete the Temp_User Windows account