What is the potential fallout of changing SQL Server's log on account? I need to be able to read files from a UNC path in TSQL and am getting an access denied error.  I've read I must change the user SQL Server logs in as to a domain user, it's currently running as NT Service\MSSQLSERVER. 

If I make the change, what potential issues should I watch out for, if any?
Changing the service account for SQL Server or changing the service account password (if you were already using a domain account) is common practice.

The NT SERVICE\MSSQLSERVER is a service account and not a formal user login in Windows. When you update the service account to a domain user account it will get linked to that service account and retain the permission required on the OS and file system.

Simply ensure any adjustment you make to the service account (change it or updating the password) is done via SQL Server Configuration Manager. This tool handles applying the permissions properly to a new service account.