How to Look Offline on Facebook
When you visit Facebook, you automatically appear in its chat system; any of your online friends can see you and begin chatting with you. This may be a pleasant distraction if you have nothing else to do, but it can be annoying if you simply wish to check your Facebook news feed and then get back to another task. If this is the case, or if you wish to hide from someone who is on Facebook, make yourself appear offline in the chat system. This allows you to read your news feed and look at pages without interruption.
Sign in to Facebook. There is no way to sign in as invisible, so you appear as available on chat for a moment.
Click the "Chat" tab in the lower-right corner of your Facebook page. Doing so displays a list of your online Facebook friends.
Click "Options" at the top of the online list of friends that just appeared. This opens a list of options for your chat session.
Click "Go Offline" at the top of this list of options. You now appear as offline even though you are still using Facebook. When you are ready to chat again, click the "Chat" option in the lower-right corner of the page to go back online.