I would like to run multiple instances of TYPO3 (a cms written in PHP)
on a single server.

You can run multiple websites in two methods: multisite or multiple

In multiple instance, each instance uses the same source files but has
its own database and private file storage area.

In multisite, you can run multiple websites in a SINGLE T3 instance
because of its multisite feature. In this configuration there is only
on database and one php environment.

I would like to know if using multiple instances requires more php
memory than a single instance running multiple websites.

I was told that the bytecache (eaccelerator) is shared between all the
installations, and the PHP interpreter runs in global mode.

Therefore mod_php or fcgid believes it is running a single php instance
and does not know of the multipple instances.

So do I need more memory for one configuration vs the other?