Does SQL Azure automatically geo-replication automatically failover?

You should read this article Designing Cloud Solutions for Disaster Recovery Using Active Geo-Replication

There are a few options on how to setup Azure for availability. So, based on your requirements and configuration, you may need to change SQL Connection String. The key in this article is to look at the "SLA". For example, in Option 2: Active-active compute with decoupled failover, the SLA is:

RTO = SQL connection change + database state change + application verification test

Notice the SQL connection change; this implies a connection string change.

Just need to configure "Failover Groups" on the SQL server on Azure, and then configure the group failover R/W listener in the application. It updates the DNS records automatically and points to the available database server in case the former is down due to an outage.

Just as an update for someone else who might come looking, this functionality is now available on SQL Azure. It's called auto-failover groups and was announced in May 2017. It is still in preview as of the time of writing - configuration is currently not supported on the portal and must be done via the API.

More information here: https://azure.microsoft.com/en-us/blog/azure-sql-database-now-supports-transparent-geographic-failover-of-multiple-databases-featuring-automatic-activation/