As another workaround, you can open Battle.net desktop, launch Warcraft 3 through it (and then quit), and then the test map functionality will work again. I think it has to do with the way the credential hand-off works from Battle.net -> Warcraft 3 (Battle.net keeps your auth tokens refreshed, whereas launching Warcraft 3 manually may use stale tokens--plus that Warcraft 3 internally doesn't "remember" your credentials for next time when you sign-in in-game). I've had the same issue with WoW for a long time so I've almost always had to re-launch the game from the Battle.net launcher.
Blizz's battle.net integration has always been pretty frustrating. And it is frustrating that it requires a connection to begin with when you're just trying to test your map.
