Harvest is an official integration - we have an API for integrating public apps, but nothing we officially support for integrating into the UI. I'm not saying it's categorically impossible, but we certainly don't officially support this. Depending on what you want to do, Greasemonkey might be feasible for personal purposes, but we'd probably discourage customers from installing hacks like this. One misbehaving extension can destroy a lot of work - not just out of malice, but by accident, especially if something in our UI changes and the assumptions the Greasemonkey script made no longer applies.
In general, we'd love to provide hooks into Asana for third-party developers in future, but it's probably a long ways off. Can I ask what you're looking to provide? It may help inform our thinking for our roadmap.