I know that "the" is wrong here, but I can't explain why it's wrong to my friend here, and I have trouble formulating a google search to find some descriptive "rule" or something. Help please?
It's like saying the date. You say:
"The 3rd of September" or "September 3"
"The September 3" or "The September 3rd"
Because what warrants the use of "the" here is that specific day of September.
In your example, what warrants the "the" is the specific number, not the fact that it's a room among other places. So you have two choices:
Either "the 24th room" or "room 24"
I hope this makes sense.
Edit: I guess time isn't a proper analogy but it carries the meaning across. Because when telling time, "September the 3rd" would have been perfectly acceptable too but "room the 24th" doesn't work.