You would need to make sure logon_trigger_login can SELECT from the table:

USE GK50LIVE;

GO

CREATE USER [logon_trigger_login] FROM LOGIN [logon_trigger_login] WITH DEFAULT_SCHEMA = [dbo];

GO

GRANT SELECT ON dbo.LimitedLogins TO logon_trigger_login;

GO

Then you could modify your trigger like this:

USE master

GO

CREATE TRIGGER connection_limit_trigger ON ALL SERVER

WITH EXECUTE AS 'logon_trigger_login'

FOR LOGON

AS

BEGIN

-- the list of logins to restrict

IF (

SELECT NUM_ALLOWED

FROM GK50LIVE.dbo.LimitedLogins

WHERE LOGIN_ID = ORIGINAL_LOGIN()

) < (

SELECT COUNT(*)

FROM sys.dm_exec_sessions

WHERE is_user_process = 1

AND original_login_name = ORIGINAL_LOGIN()

)

BEGIN

ROLLBACK

END

END;

GO