Is ASYNC_NETWORK_IO wait type anything to worry about?
As you said, this wait type indicates that the application is not keeping up with SQL Server. Now what that really means is, that SQL Server can't send the data over the network as fast as it would like.

There can be two underlying causes:


The app is written inefficiently and does not process the rows fast enough.
The network is maxed out.


If the application itself is too slow there will be no or no significant impact on the performance of other queries. If on the other hand the pipe is too small, other queries can't send their results either and have to wait.

In the latter case however you would have all connections waiting on ASYNC_NETWORK_IO. You should be able to see that impact clearly.