Turn off MacBook Air screen rendering but maintain running applications? I would like to completely turn off the screen rendering on my Mac for some time as I have some processes running in the background but I don't need to use the monitor. I'm expecting that this measure will save some battery consumption, but that's beyond the goal of this post.

I've been using Control + Shift + Fn + Power Button to sleep the display. The problem is that the background processes stop running.

How can I keep my background processes running but display nothing on the screen?

I'm using a Macbook Air if it makes a difference.
I could be missing something in your post, but you should be able to set your Energy Saver preferences so that your display goes to sleep, but your computer doesn't.

Since you haven't specified the exact version of macOS you're running I can't give you step by step instructions or include a screenshot, but essentially you need to go to Apple > System Preferences > Energy Saver.

If you take a look there you should see you can set your display to go to sleep at a certain point and your computer to go to sleep at another point in time (or never, if that's what you need).

Doing so will save you on battery power as your display is asleep, but of course your MBA is still running otherwise.

If you're going to have your MBA plugged into an AC power source when you need the processes to run you'll need to ensure your settings are configured on the Power Adapter tab (as opposed to the Battery tab). Likewise, you'll need to set your Battery tab settings appropriately if you're also going to be running these process while running off the battery.

You'll also see other settings here (depending on your Mac and version of macOS), so you may want to familiarise yourself with the Use the Energy Saver settings on your Mac page on Apple's website.