<p>I downloaded a VM image of a web application that uses MySQL.</p>

<p>How can I monitor its space consumption and know when additional space must be added?</p>

If only MySQL is available, use the SHOW TABLE STATUS command, and look at the Data_length column for each table, which is in bytes.
If you have other languages available on the machine, a script in any of them that runs regularly (cron), checks disk free space or size of database directory, and updates you over e-mail or otherwise. There are far too many options to suggest a particular solution -- it depends on your situation.