Thanks cullenl for leading me on the right path. I don't have an answer as to what's going on but I did find a good starting point.
I compared the /etc/asterisk folder for two identical installations, one that crashes and the other that does not. Both servers are running identical versions of FreePBX and Asterisk. What I did find was that in the functional server, there are symlinks which are missing in the crashed server.
All of the following are missing in the server that crashes:
lrwxrwxrwx 51 ari.conf -> /var/www/html/admin/modules/arimanager/etc/ari.conf
-rw-rw-r--. 766 chan_dahdi.conf
lrwxrwxrwx 59 confbridge.conf -> /var/www/html/admin/modules/conferences/etc/confbridge.conf
drwxr-xr-x. 4096 keys
lrwxrwxrwx 52 logger.conf -> /var/www/html/admin/modules/logfiles/etc/logger.conf
lrwxrwxrwx 54 queuerules.conf -> /var/www/html/admin/modules/vqplus/etc/queuerules.conf
lrwxrwxrwx 67 res_digium_phone.conf -> /var/www/html/admin/modules/digium_phones/etc/res_digium_phone.conf