Go into your minecraft folder and open up options.txt .
Find the line that reads pauseOnLostFocus:true and change true to false .
Now Minecraft won't pause to the menu screen when you tab away.
You can find your minecraft folder in a couple ways: Using the file system:
Navigate to... %
appdata%\.minecraft\ (on Windows)
~/Library
/Application Support/minecraft/ (on Mac OS X)
~/.minecraft/ (on Linux) Using Minecraft's menus: Start Minecraft and click Options , Resource Packs , and then Open resource packs folder .
In the file system window that opens, go up one folder and you'll be in your minecraft folder.
gaming.stackexchange.com/users/54182/user54182
You can press F3 + P to let the game not auto-pausing.
Press F3 + P again to re-enable it.
gaming.stackexchange.com/users/80773/infmagic2047
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.
gaming.stackexchange.com/users/4775/Dave McClelland
Since the update that merged the single- and multi-player code bases, you can press T to start typing in chat, which will let you tab out without pausing or hiding the game .
gaming.stackexchange.com/users/24394/Tharwen
This answer is no longer relevant to current Minecraft versions, because F1 no longer hides menus; also, F1 is now a toggle and the tricks for holding it down are irrelevant.
Since you mention that your concern is that you can't see (as opposed to having the menu up at all), I have a fix for that:
with the menu up, use F1 to hide it.
Pressing F1 removes all HUD elements and leaves only the game view.
However, you have to hold down F1 .
There are several tricks for this to fool the game into thinking you have F1 down: Press F1 and switch out of Minecraft while holding it down.
(Tested on Mac) Before releasing F1 , press the Alt (Windows (and Linux?)) or ⌘ Command (Mac) keys.
Mac-specific: If you have a system where you have to press fn - F1 , just release F1 before fn .
gaming.stackexchange.com/users/5962/Kevin Reid
Whilst all the other solutions look perfectly fine, I thought I'd add the method I use, when I want to not have the game pause, I hit esc, and "Open up to LAN".
The game doesn;t pause then on focus loss.
gaming.stackexchange.com/users/61520/TMH
here's how I did it, I was mining, in a map called Minevolution, I was in full screen mode and then, while holding down left click, I pressed F11 at the same time I let go of the left click button, and it continued to mine, then I just did the whole Alt+Tab thing to get out of it
and it worked
gaming.stackexchange.com/users/93595/Reo Reyes
If you just open a LAN connection the game no longer can be paused, even if you're in the game menu.
A LAN connection only allows people to jump in if they are on the same internet connection, so it is pretty secure.
gaming.stackexchange.com/users/100610/user100610
Is there a way to alt-tab out of Minecraft (i.e. make Minecraft lose focus) without the game auto-pausing and without having to bring a menu (e.g. inventory) up first?
Can I alt-tab out of Minecraft without the game auto-pausing?
gaming.stackexchange.com
<minecraft-java-edition>
gaming.stackexchange.com/questions/15664
gaming.stackexchange.com/users/2616/Brian Beckett
Try F3+P to remove auto-pause. You can also open a LAN connection. There are multiple ways you can do this, chief among them being editing the options.txt in the minecraft folder. There is also a key combination that works (F3 +P), or keeping the chat box open. Opening a LAN connection also works. 
Users say that opening the minecraft folder and changing an option in a text file works. A key combination also works. Keeping the chat opened when tabbing out also works. Opening a LAN connection also allows this.