I'm not familiar with a way to actually remove the game from Origin, but there is a way to hide the game from actually appearing in Origin. Open the file
C:\ProgramData\Origin\InstalledGames.xml
The Battlefield Beta has an ID of 71171. Find that line and change
hidden="0" to hidden="1"
I'm not quite sure what the ID for the alpha is, but you could just follow a similar process to remove that one as well.
I know of only one way that this can happen in windows, and has nothing to do with Steam other than their not-so-great decision to install all content into Program Files.
If you are on Windows 7 or Windows Vista, and you have UAC set to ON, then change it later to OFF, this can happen.
UAC does thing evil thing where if a program tries to write to a "protected" area like Program Files, EVEN IF YOU ARE ADMINISTRATOR, when you get a UAC prompt and give permission, it actually writes any files that would normally go into that directory into a "virtualized" directory somewhere else.
If you turn off UAC, that will no longer happen, and as an awesome bonus, it will no longer know about the files that it virtualized.
So in your case, if you had UAC on, installed a ton of games, then later got sick of UAC and turned it off, this would happen as you described.
See the second paragraph in "Features" in the article on User Account Control.
If you have done this and turn on UAC again, your content will come back, but be a TOTAL mess because if you've downloaded more stuff with UAC off, then that will be invisible when you turn UAC on, and vice versa.
this drove me a little crazy once before I realized what was happening.
i'm curious to know if this is your issue.
Best Answer
Apparently, the game's process is still running in the background but somehow crashed (or something else. The thing is that it decided to take a nap in a corner of your computer). The only solution is to force the process to stop.
Open the Task manager (use the shortcut CTRL+SHIFT+ESCAPE) and open the process tab. Find the process for the game and stop it (you can just select it and press the delete key on your keyboard if you are too lazy to click the button :P I can tell you I am too lazy to click it!).
Once this has been done, try launching the game again. if it doesn't works, close Origin and follow the same steps.
If it still doesn't work, restart your computer.
If all else fails, contact EA Support.