There is no direct way to publish from Lightroom to Instagram however, you can look for third party plugins for the same, if available.
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On Windows, I use Lightroom to export the photos to the hard disk, and from there I use third-party software called Gramblr to send the photos to Instagram.
Many thanks, Chetna.
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There are many third party utilities for Mac OS X and windows that allow you to upload directly from your desktop however instagram tends to revoke their API keys because they really really want you to only use mobile platforms. This is also why there is no plugin for Lightroom to do this and why IG doesn't simply allow upload in their web interface. Currently, the simplest way to get images on instagram using Lightroom native tools is to put those images you want on IG in a synced collection. Sync them to mobile and post to IG from Lightroom mobile on your iOS or Android device. This works quite well but is not as full featured and simple as the desktop apps.