As far as I can tell, scaling an Azure DB from the retired tiers to the new tiers is simply a matter of using the scale function in the Azure portal.
What I cannot seem to find anywhere is a definitive answer as to whether there are any connection string changes required (or any other issues that could cause unavailability) when scaling from the retired to new tiers.
I have a production database that needs to be upgraded, service interruption would be very bad.
The scale operation will not change the connection string. You could face some (very small, but) finite amount of downtime while the switchover happens.
Please refer to the documentation for details. Note that you will be have to suspend geo-replication (if already enabled) for the duration of the upgrade.