Why does Apache fail when the partition is full?
- Yes. Disable logging or get logrotate configured correctly to rotate and/or compress your log files.
- If log files are activated, the system has to assume that you want logging and it's a terribly bad idea for any server application to continue working when it can't log what it is doing. It's quite likely that the person who configured the logging needs it for one purpose or another.
Why is your disk full anyway?
You seem to have multiple problems.
- It's possible that your server disk gets full. Don't you have any kind of trending and graphing (Cacti, snmp+mrtg, Munin) and monitoring (Nagios, Monit) in place? If you don't get notified about a disk getting more full than some % threshold, I bet you won't get alerts about other serious issues, either.
- You don't seem to have a sensible log rotation scheme or central logging configured.
Fix those. And if you have a busy server, configuring Apache to use syslog facilities instead of Apache's own log files can actually increase the performance, as Apache will not wait for the log entry to be actually written, it merely throws it at syslog trusting the message will get logged.
Google how you can use Apache's CustomLog directive together with logger and how you can set up a centralized log server. It will be good for you in the long run.