Automatic recovery on a 3 node AlwaysOn Multisite Cluster?
The simplest answer is, no, you won't be able to facilitate this with a 3 node cluster in the manner described.

The reason is due to quorum. Assuming the 3 nodes, 2 at Primary and 1 at DR with Windows Server 2012R2. Dynamic quorum is on by default, this will automatically adjust node weights in case of a node failure. Dynamic witness is also on by default which will change the witness vote to keep the number of total votes odd.

The thing is that dynamic quorum only works if less than half of the nodes go down simultaneously. If 50% or more voting nodes go down at once there won't be enough voters left to keep quorum or for dynamic quorum to decide that this isn't a split brain scenario.

How could you potentially achieve this?

If it would be possible to put 2 nodes at the Primary site, 2 nodes at the DR site, and a witness at a 3rd site then it should do what you're looking for.


  is there any way to configure this in a way which automatic failover will occur if the primary and DR sites lose connectivity or if the primary site goes down entirely?


These look the same from the perspective of the DR site. Whether it loses connection with the servers at the primary site or whether the primary site goes down doesn't look any different. In each case they no longer can "see" the other nodes, only the local ones. This results in the race to acquire the lock on the witness. Whichever side attains the lock first, wins.

There is an additional setting in Windows Server 2012R2 called LowerQuorumPriorityNodeID which can be used to weight one side or the other when these types of situations happen.