External drive mounts but doesn't appear in Finder anymore
With the help of Drobo's technical support who pointed me to this MacRumors thread, I've resolved the issue. It turned out that somehow my drive got set to "invisible", which is a filesystem attribute. I don't think this setting is settable via the Finder, so I'm not sure how it got set like that.

To confirm the issue, I had to install Xcode and then install the command line tools, then run this Terminal command:

GetFileInfo /Volumes/Drobo/


where /Volumes/Drobo/ is the path to the volume in question. This command returns all the attributes set for that volume. If the list of attributes includes a V (capital V), it means that the volume is invisible.

To fix it:

SetFile -a v /Volumes/Drobo/


with a lowercase v, to remove the invisible attribute.

After a reboot, everything was back to normal.

Note: depending on the drive and your permissions, it's possible you will need to run these commands with sudo, although I didn't.