Maybe this is more ammunition for people who think that RSS is dead.
Since its early days, Tumblr has offered a useful import feed feature to help you bring content from other sites – like Flickr, Twitter, YouTube, Facebook, Blogger, etc. – into your main Tumblr blog. Technically, you could import content from just about any external site into Tumblr as long as there was an RSS feed available.
Unfortunately, as @Shripriya first noticed, Tumblr has quietly dropped support for importing RSS feeds into their system. Tumblr’s support site still has a page on how to use Import feed but the feature itself has been removed, thus denying you the ability to use Tumblr has a lifestreaming service.
The last auto-fetch request into my Tumblr blog was made during the first week of June and that may be the date when they dropped support for RSS imports (you can still export your Tumblr blog as an RSS feed though).
I am little surprised at Tumblr’s approach here but that doesn’t mean you can’t bring your external content into Tumblr – they do offer a simple API that can write anything, including RSS feeds, to a Tumblr blog but you’ll have to take the help of a developer to get that thing working for your blog.