PHP/APC fatal error, apc_mmap: mmap failed

If this does not cause visible errors on the site, it must happen so early during the apache child process creation that no request has been transferred to the child before the error. Googleing for the error message suggests (http://www.litespeedtech.com/support/forum/showthread.php?t=4242) that your APC configuration is incorrect and you should check following configuration parameters:

apc.enabled = 1
apc.shm_size = 48 ;; MB, make sure that this is allowed by the system
apc.include_once_override = 1 ;; test both values
apc.mmap_file_mask = /tmp/apc.XXXXXX ;; verify mask if you use file backed storage

I'd guess some APC process is requesting more SHM mmap'ed memory than the kernel or system config is willing to provide.