The general gallery seems to work. The individual image display seems to work. Here are my test results:
* [flickr-gallery] - seems to work fine.
* [flickr]...[/flickr] - seems to work fine, although if you have any spaces at the end before [/flickr] it breaks.
NOTE - quotes in all of the examples do not work. The quotes must be removed for the attributes to be parsed correctly. This appears to be a bug in WordPress itself. All tests were done without the quotes.
* [flickr-gallery mode="photoset" photoset="72157605870230826"] - No photosets are returned. It appears that the "photosets_getPhotos" function in phpFlickr never even gets called.
The other modes appear to work, although I didn't test them all. The photoset mode is the one in particular that is clearly not working.