I couldn't figure out how to get the Flick Blog This feature to work with my Movable Type blog. I finally figured it out last week. For those of you who can't get the Blog This feature to work in Flickr with your Movable Type blog, here is some help. I'm using Movable Type 3.33.
Log in to your MT account
Click your name (next to "Welcome" at the top of the screen)
Scroll down until you see the Web Services Password field and enter a password different from the one you use to log in to MT.
Now go to Flickr
Click the You tab and select Your Account
Click Extending Flickr
Select Blogs
Enter your blog's CGI directory and then your login info. Be sure to enter the web services password you just selected.
Save it and you can start blogging directly from Flickr.
I really hope this helps someone. I saw a lot of posts on the Flickr forums about people having this problem and few were able to solve it.