How long can E.coli last in test tubes of Terrific Broth in a 37 Celsius incubator?

I remember E. Coli living for hours not days in a lab class. What does your lab instruction sheet say?

This is for work, not a lab class, by the way. I mean, my boss showed me how frozen E. coli can still be cultured from like four years ago, and I did just that and they grew on the plates and then I transferred them to sterile test tubes overnight to grow. But I seem to have run out of time. My boss is currently out at the moment.

At 37* C they will run out of nutrients in hours. Your work instruction should give you a timeline to complete the project.

E. coli divide rapidly at 37C, and need to be shaken vigorously if stored at this temperature otherwise they will consume their surrounding nutrients faster than diffusion can supply them from the bulk media. You will end up with a pellet of very dead cells at the bottom of your tube by tomorrow morning (probably by midnight tonight). If you really cannot get to them until Monday, you can store the tubes at 4C (in the fridge) or, mix the culture 1:1 with 7% DMSO in PBS and freeze at -80C. Just scrape a small amount of the frozen culture into a new tube on Monday.
As far as your plate question, bacteria will survive quite a long time on plates, but there are hazards to your experiment if you let them sit too long. Namely, depletion of your selection antibiotic and the growth of non-plasmid carrying cells, aka “satellite colonies”. What I do if I am too busy to deal with them is grow them to a visible single colony state, usually overnight does it if you streak at 3 or 4 PM, and then move the plate to the fridge. The bugs will be OK for about a week before you start having problems (though I have gotten away with significantly longer).