If you are just running the software on a server somewhere, then no, you don’t need a commercial license (this may be a provision in GPLv3, but MySQL isn’t licensed under the GPLv3).
If you’re going to redistribute the software, then you will need one.
If the application doesn’t require MySQL, but makes use of it if present (perhaps MySQL is one of several options, for example), and so doesn’t depend upon the MySQL client library, and doesn’t statically link with it, then you may be able to do without the commercial license (you will want to check with a specialist lawyer, though, which may cost you more than the cost of the license!)
There are also two alternative options: use an alternative client library distributed under the terms of a different license, or write your own. There are several alternative libraries around (PHP includes one of its own, as an alternative to the module built on top of
libmysqlclient for example).