Redirect all logs to a network share - local HDD "disappears"
I don't believe to you can tell OS X to write the logs to a different location BUT you can move the location to which it writes!

Not sure what logs you are interested in, but lets say it's /var/log.

1 Copy the /var/log directory to your remote share. 

2 Rename original /var/log to something else 

3 Create a symbolic link from /var/log to the remote share.

Now when the system tries to write to /var/log it will really be writing to /yourshare/whatever/var/log

Have not tried this myself but should work.  

Some other ideas from the linux world, here:

https://serverfault.com/questions/55984/how-can-i-move-var-log-directory