In more recent versions of ESXi the profile has required that you configure persistent storage and specify a coredump file in the Advanced Settings of the configuration (Advanced Settings>>Syslog>>global>>Syslog.global.logDir). I've found this to be very finicky. You have to make sure that you specify the proper datastore in the syslog settings in the proper way. The example they give you for how to type it in barely seems to ever work. I always type it in like this: [datastore02] logdir
Usually that takes care of it for me, but today I ran into an issue where the error telling me I didn't have a coredump target specified wouldn't go away. It would accept my datastore and file without error and retain the information, but the error just wouldn't go away. I tried all sorts of crafty tricks to make it go away. In the end I simply vMotioned the servers off that server and restarted the physical host. Of course, that's what ended up taking care of it.
FYI: This was with ESXi 5.5