Old versions of replays runs an old version of Starcraft 2 to display, when this is run you are logged out of battle net and there for no login is requred to watch replays of non current versions games, once the replay is done you are either logged back in or brought to the log in screen depending on where you where when you started the replay, also all social features associated with battle.net is turned off when watching old replays source
Not sure if it is possible to watch replays for current version with out logging in because the game wants the social features turned on, but you can watch replays with a starter edition account. source
If you don't want to install the game you can not see the replay, but the best solution i could find was sc2gears, i haven't tested it but it looks ok.
only mention of a tool that enables you to watch replays was from beta, not sure if it has been updated and it also edited game files so would be against tos to use it.