The article doesn't explain why this is a bad idea for database servers hosted on a remote machine. The first reason is obvious... Network connections take memory and processing power on both client and server. Each additional query causes more resource usage. It is unnecessary overhead which is why things like multiple active result sets were created for SQL Server.

The network round trip time can also add up if you run into resource constraints doing this.

On a remote database, you also have to contend with multiple users and so complicated locking techniques can come into play depending upon the complexity of all database activity.

Many databases have options to return multiple result sets from one connection which helps control the overhead caused by this usage pattern.

EDIT: This also brings back horrible memories where developers would do this in a db client server architecture. Then they would often not close the DB connections when done. So you could have thousands of active connections basically doing nothing. Luckily, this problem was solved with better database connection handling.