LÖVE

Say one of my threads gets caught up in an infinite loop. I know this because I expected it to callback but it never did after a few seconds. All I want is to retrieve a table containing some important data to understand what was going on. Is there a way to do this? It's all hypothetical so please tell me if there's a flaw in my mental model.

No. At best you can insert some code that dumps that data in some place you can access later, like the console or a log file. Using something like GDB would mean you need to be familiar with LuaJIT's internals, and even then, it would be a huge pain to decode the full contents of the table without some kind of specialized debugger to interpret it.