"Sorry, it appears the EA Servers are currently down. We are working to address

@optycalillusion @ChequeMateX

I just checked and I can logon to the ME3 servers just fine, so either the problem is fixed or it's a regional issue. 

Anyway, you can play the Single Player Game without server connection. Either set Origin to offline mode, or if that don't works disable your internet connection completely. (Pull the plug or disable the adapter) Then the game will run with the installed DLCs enabled.