As of the latest update to Steam, you can now PERMANENTLY remove games from your Steam account!

Steps

1. Log into Steam Support
2. Select Games, Software, etc.,
3. Select the game from the search function that shows up
4. Select the option: I want to permanently remove this game from my account.
5. Select the option: Ok, remove the listed games from my account permanently.
6. Repeat as many times as you have games you'd like to get rid of.

You can browse over your entire library and choose which software you'd like to permanently get rid of. But note, once you have removed a game from your library, it's gone permanently, so if later down the road you'd like it back you'll have to purchase it again.

You'll also need to watch out for some other issues. The thread linked above has noted that if the game is part of a bundle (HumbleBundle, etc.), removing one game from a "package" may require you to remove all the games from that specific bundle.

From helping others on Steam support forums and their result from following my suggestion, I have concluded the following as the cause:

Steam exits in an un-friendly way, thus not syncing your shortcuts with the cloud. This can occur when you let it (Steam) close by itself when shutting down/restarting your computer, Steam basically force closes, it does not close gracefully.

The main reason this is the cause is because Steam syncs your shortcuts with the cloud, but in order to do this it must shut down "gracefully". This means you have to shutdown Steam by doing:

Steam > Exit or Right-Click on Steam Icon Taskbar > Exit

Either of those will allow Steam to sync with the cloud if necessary.

Others say it is due to corruption of a local file, but it can be fixed by doing what I have suggested above. There hasn't been an official response from Steam on this issue, so it's just a logical guess based on past experiences of myself and others.