Can the 2009 Mac mini work with its optical drive removed?
No it does not. Many have swapped out their internal DVD-ROMs on favor of installing another bay that can handle an HDD/SSD.

Here's an example of an optical to HDD bay: http://www.ifixit.com/Apple-Parts/12-7-mm-SATA-Optical-Bay-SATA-Hard-Drive-Enclosure/IF107-081