Minecraft closes whenever I alt tab from full screen

For single player, it seems that you've already figured out that bringing up the inventory screen prevents the game from auto-pausing when it loses focus. I don't believe there is any other way to prevent the auto pausing.

You may, however, consider hosting a SMP server and being the only person connected to it. In SMP, the game will not pause when you are alt-tabbed out. You can install mods on a SMP server that allow you to create user whitelists to define who you want to connect to your server. If you add only yourself to that list, you don't need to worry about anyone else connecting while you are playing.

Also, you can use the escape button to exit out of any gui, including any container inventory. I've always used this, so I don't know why Minecraft allowed us to close the container too! (If that makes sense)