I've been having the same problem as others for the last couple weeks. I enter my credentials, specify a blog to import, then wait. And wait. A week goes by, and I figure *something* should've happened by now.
I found this thread with the tip about turning off the custom domain for the Tumblr blog you want to import, but I've run into what seems to be a new problem.
Today I cleared my Tumblr credentials from the importer to start fresh. I was about to shut off my custom domain when I found that the importer can't even connect to Tumblr anymore. I get "Tumblr replied with an error:" but, of course, the error is blank.
I tried emailing Tumblr about this and I'm not entirely sure the rep who replied understood what I was asking. But if they did, they made it sound like WordPress' Tumblr importer isn't supported by their API.
Anyone know what's up? Has there been an official word from WordPress about what's going on? Any ideas would be much appreciated.