I had a customer where we had installed Veeam backup and configured it to take backup of a SQL server,after some days the customer contacted med, that one of the log drives on the SQL was running full.
The backup job was running with no errors and in job could we see that it had done the transaction log truncing. but that was not true.
We had enabled Application-Aware backup option i the job, and the user is a member of the local administrator of the virtual machine, but in the event log, I could see that it failed to login to the SQL server, so vi decided to make the backup account member of the sysadmin account, witch is also is necessary for Veeam to identify the SQL databases. But it also seems that it needs it for truncation of logs.
This is different from older version, see Veeam KB: http://www.veeam.com/kb1746
UPDATE (July 17th 2018)
Veeam has made some changes in 9.5 update 3a, to the required permissions for SQL log truncation and log backup.
The service account used for the application aware process, still needs to be a member of sysadmin, or have the permissions that is outlined i the documentation here: https://helpcenter.veeam.com/docs/backup/explorers/vesql_permissions.html?ver=95
But they have changed the logon behavior of the application-aware application, it now don't uses interactive logon, for security reasons. Instead it uses "logon as a batch job".
This requires that the service account should be added to the local security setting "Logon as a batch job", this can be done locally or thru a Active Directory group policy (GPO).
Veeam also have a hotfix that can make the application aware poccess to do a fallback to "interactive logon", but I think that you still should implement the "Logon as a batch job" solution, since the hotfix may not be included in next update.
Here is the Veeam KB about this, including the hotfix: https://www.veeam.com/kb2689