OS: Zorin OS 16 (Ubuntu 20.4-based distro)
I downloaded the Linux installer, ran it and it looked like the game installed fine. But when I want to launch it, nothing happens. If I launch start.sh from the terminal, I get a message saying "There is no data folder".

What do I need to do the ensure the game is installed completely and runs?

I downloaded the installer file, chmodded it to be executable, launched the script from the terminal, installed it without an error message, then tried to launch it.
It is installed in my home dir, under GOG Games/Immortal_Redneck/ (but I also tried in GOG_Games/Immortal_Redneck so as to not have any spaces, still the same result).

Here's what I get when I try to run the game.

istvan@GSzob:~/GOG Games/Immortal_Redneck$ ./start.sh
Running Immortal Redneck
Set current directory to /home/istvan/GOG Games/Immortal_Redneck/game
Found path: /home/istvan/GOG Games/Immortal_Redneck/game/ImmortalRedneck.x64
There is no data folder

This is very strange.
I made a research on the web and found out that this seems to be a bug that affects games based on Unity. According to this discussion, the bug apparently appears when the game is installed on particular types of partitions.

I would try reinstalling the game into a directory with a path as simple as possible (try using lowercase letters only, without special characters and spaces), such as "~/gog/redneck".

If this doesn't work, I'm afraid you should try to install the game into a different partition.
If there isn't another partition with enough disk space in our system, you can create a dummy one using loop device as suggested here. If you need more information about loop devices, you can go here or here.

Thanks for digging that up. Now I installed it on my / partition, under /GOG/IR and it launches; strange, / and /home are both ext4, only /home is much larger. Must be something with the size, as pointed out in that thread.

Good. Enjoy the game. Just out of curiosity, have you tried installing it in the /home partition in a simple path using lowercase letters only, as I suggested?

I think I did, under ~/gog/ir, but I wouldn't swear on it; I know I planned to but between all the reinstalls, I can't remember 100% if I got to it and it didn't work or if I went to /GOG/IR at once after all.
EDIT: I tried now, no, it does not work in ~/gog/ir.

You weren't supposed to try this just to satisfy my curiosity. :-)

Anyway this might help others in the future: this game, or maybe Unity in general, has definitely some serious problem with big partitions.