Upgrading and testing module updates one at a time, I came across the problem.

When updating imagecache from 6.x-2.0-beta10 to 6.x-2.0-rc1, the following error was created:

Parameter 1 to imageapi_gd_image_resize() expected to be a reference, value given in /drupalroot/sites/all/modules/imageapi/imageapi.module on line 165.

The version of imageapi that I was running was 6.x-1.6. An upgrade to the latest 6.x-1.10 solved the problem.

There's little that can be done in code to solve the problem as the ability to add version dependencies to the .info file is not available in D6. But upgrading to the latest version of imageapi should be noted as a requirement in the troubleshooting guide, at the very least.