You need an Internet connection to initially download content. Once Modern Warfare Remastered has been downloaded, you do not need to be connected to the Internet to play/launch the Single Player Campaign or play Multiplayer locally. Will Call of Duty: Modern Warfare Remastered come as a separate disc?
Yes, you can play both the campaign mode and multiplayer mode in Call of Duty: Modern Warfare 3 offline. The campaign mode can be played in its entirety in offline mode, and a Private Multiplayer mode allows you to access all of the maps and modes present in Multiplayer.