UPDATE: As of further investigation I have found this works for all 'Press-and-Hold' keys, such as WASD and LShift.

When you jump and fullscreen at the same time, you start constantly jumping; and move as well, if a movement key was pressed. If you enter chat mode, jump, or press escape, it stops. Moving and looking around does not change the situation, apart from forward movement changing to up and down when you walk around. I have tried this on 2 accounts, 2 computers and 2 operating systems. A video is attached which show the screen and keyboard (So you know I'm not pressing space). It works in singleplayer and multiplayer and in creative, only seems to work when jumping off a elevated section, otherwise it just travels forward. You can see the jumping in third person and you can see others also jumping. I don't have enough computers to test other operating systems but the problem is on Windows 8.1 and Windows 7.