If you export from Adobe Lightroom to Dropbox sync folder, then queue a new photo post on Tumblr.
In Lightroom create a "hard drive" share service and set the export location to a folder the Dropbox software synchronizes. When the photo is saved to Dropbox's online server, it's queued to post (useful if you export a bunch at once so you don't flood your followers' feeds). I use this because I have a watermark for my photos specific for Tumblr. Make sure your share service renames the exported files to the exif title if you name your photos.