Can the DB configuration be set up so that if one DB fails it will automatically reconnect to another copy of the DB? Or is something like that usually managed with a load balancer and/or virtual IP?

This it not handled directly by ABIS.

This has to be configured on DB side. Every enterprise SQL DB engine supports kind of high-availability mode (e.g., Oracle has Oracle RAC). 

After you have running DB HA cluster, correct connection string has to be set in ABIS configuration files.

It depends on particular DB whether there is one cluster's IP address or multiple addresses in JDBC connection string. 

Production verified scenario is a High Available database with 3 nodes and floating (virtual) IP. ABIS is connected to a virtual floating IP and DB nodes behind it could switch over an active node as per the real life needs.

Relevant ABIS version: Any