Why does some polyurethane need to be wiped off, and what happens if I don't?
There's a chance you were using wiping varnish or a "tung oil finish" (not really tung oil). It's basically thinned down poly that can be left on in a thick coat, or the excess can be wiped off after a coat is applied.

The warning to not let the finish dry without wiping is probably good advice to eliminate the possibility of runs and drips.

Popular Woodworking has a good article on the history of wiping varnish.