It means "only" or "simply".
However, the context is important. Here is a typical case where an imperative sentence is used with "just":
Dad: Jimmy, mow the lawn.
Jimmy: But daaaaaaaaaaad, I don't want to mow the lawn!
Dad: I'll get ice cream if you mow the lawn.
Jimmy: But I'm playing a game!
Dad: Mow the lawn now or I'll give you time out!
Jimmy: No! Please don't!
Dad: Stop complaining and just mow the lawn!
In this argument, it means: The only thing you should do is mow the lawn. You should not complain, or argue, or do anything else. Obviously, it also conveys anger or frustration.
Compare with a more common use of "just":
Do you want ketchup or mayo or garlic yoghurt?
Just ketchup, thanks.
Here it means: The only thing I want is ketchup. I do not want mayo, or garlic yoghurt, or anything else.
In the argument context, it can also be seen as "simply". "Simply mow the lawn, without extra complications like complaining or arguing."
The shirt is similar to the argument, except much shorter, and perhaps with less anger and more frustration.
Dad: Don't open the door.
Jimmy: But daaaad, I want to play outside!
Dad: Just don't open it!
Mad scientist: Don't press the button.
Jimmy: But it's shiny and red! I want to press it!
Mad scientist: Just don't press it!
In this case, nobody is complaining about being told "do not disturb" - obviously, since it's a T-shirt and not a two-way conversation. The writer is acting as if someone did, anyway. It's like a pre-emptive answer, because the writer is feeling especially frustrated and is expecting someone to complain:
Mom: Don't eat the cookies. Just don't!
Jimmy: But -
Mom: I said, "just don't!" Last time you ate all the cookies and we had to cancel the family picnic!