It is a good idea to save the database server's log output somewhere, rather than just discarding it via /dev/null. The log output is invaluable when diagnosing problems. However, the log output tends to be voluminous (especially at higher debug levels) so you won't want to save it indefinitely. You need to rotate the log files so that new log files are started and old ones removed after a reasonable period of time.
If you simply direct the stderr of postgres into a file, you will have log output, but the only way to truncate the log file is to stop and restart the server. This might be acceptable if you are using PostgreSQL in a development environment, but few production servers would find this behavior acceptable.
A better approach is to send the server's stderr output to some type of log rotation program. There is a built-in log rotation facility, which you can use by setting the configuration parameter logging_collector to true in postgresql.conf. The control parameters for this program are described in Section 19.8.1. You can also use this approach to capture the log data in machine readable CSV (comma-separated values) format.
Alternatively, you might prefer to use an external log rotation program if you have one that you are already using with other server software. For example, the rotatelogs tool included in the Apache distribution can be used with PostgreSQL. To do this, just pipe the server's stderr output to the desired program. If you start the server with pg_ctl, then stderr is already redirected to stdout, so you just need a pipe command, for example:
pg_ctl start | rotatelogs /var/log/pgsql_log 86400
Another production-grade approach to managing log output is to send it to syslog and let syslog deal with file rotation. To do this, set the configuration parameter log_destination to syslog (to log to syslog only) in postgresql.conf. Then you can send a SIGHUP signal to the syslog daemon whenever you want to force it to start writing a new log file. If you want to automate log rotation, the logrotate program can be configured to work with log files from syslog.
On many systems, however, syslog is not very reliable, particularly with large log messages; it might truncate or drop messages just when you need them the most. Also, on Linux, syslog will flush each message to disk, yielding poor performance. (You can use a "-" at the start of the file name in the syslog configuration file to disable syncing.)
Note that all the solutions described above take care of starting new log files at configurable intervals, but they do not handle deletion of old, no-longer-useful log files. You will probably want to set up a batch job to periodically delete old log files. Another possibility is to configure the rotation program so that old log files are overwritten cyclically.
pgBadger is an external project that does sophisticated log file analysis. check_postgres provides Nagios alerts when important messages appear in the log files, as well as detection of many other extraordinary conditions.