How do I remove a Steam game from my library?

Permanent Solutions

Permanently removing game(s) from library

Valve has recently added the ability to remove games manually, without the intervention of the support team. See @skully's answer below for more details.

Contacting Steam Support

I've just been through this. Steam Support is really good at sending canned responses and not great at actually doing anything. The winning argument for me was to quote the Steam subscriber agreement (my emphasis):

10 TERM AND TERMINATION

...

B. Termination by You.

You may cancel your Account at any time. You may cease use of a Subscription at any time or, if you choose, you may request that we terminate your access to a Subscription. However, Subscriptions are not transferable, and even if your access to a Subscription for a particular game or application is terminated, the original activation key will not be able to be registered to any other account, even if the game or application was purchased in a retail store....

I was able to convince them to remove games I no longer wanted with a couple of exceptions where the game was part of a bundle and I wanted to keep other games from the same bundle.

FWIW The first canned response I got was identical to the OP's. If you contact Steam Support you can still have them remove the games from your library as opposed to the other solutions here.

@lunboks pointed to a handy link in the comments below which explains how to use the steam console to find out where a particular game came from (what subscription/bundle/etc).

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Workarounds

Hiding the game in your library

A previous Steam Update includes the following feature:

Add "Hide this game in my library" functionality, accessible from the Set Categories menu. Games that are hidden will not show up in library filters except for a new filter called "Hidden", which will only appear once at least one game is marked as hidden. Hidden games are still available for play and will still appear on the user's profile.

The game is hidden from tile view, search results, etc.

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.

Changing the visibility

As Mr Smooth mentioned, you can use the drop down menu in the library, to only display installed games.

Adding categories

You can also add categories, and add the game(s) to the category, this can be done by right clicking a game, and selecting the "Set category..." button. From there you can add categories.

Contacting Steam support

You can also try to contact Steam support, about removing a game from your library. They should be able to do it. Please note that this may take a bit of arguing, and will require more than one e-mail exchange.