This one is almost funny, because it is so dumb.
So, I open the Battle.net launcher, and launch the World Editor from there. I open my map and click on “Test Map”.
And somehow, there is a connection error (why in the wildest F would I need a connection to test a map) and I get thrown into the menu where you can choose to play offline, and even the “Play Offline” option is greyed out.
I can’t even… Damn you, Blizz.
Same issue.
Edit: Not sure if this was OP’s issue or not, but I resolved my problem by starting the Warcraft III client normally, logging out, and then logging back in via the menu options. This seemed to store the login credentials somewhere properly. Simply putting them in each time I was prompted for them upon starting a test game did not.
I assume it is their DRM getting in the way. So, if I haven’t logged in regularly and then decide to test a map, it shoves me out because it is not authenticated. Blasted new age “asset protection/games as a service” bullcrap --’ like this would ever stop anyone from pirating the game. It just hinders whomever tries to play it faithfully, instead.