Early Beanstalk AMIs were not rotating logs properly. You can fix it by using the latest AMI in your deployment. Go to EC2 Console, AMIs. Filter the list by choosing Amazon Images, "elasticbeanstalk" then sort by "Source" to see the latest AMIs.
Alternatively, you can edit the file /etc/logrotate.conf.elasticbeanstalk on the Beanstalk server to fix log rotation. The following config appends the timestamp after file names. It produces logs like tail_catalina.log-1322236861.gz, tail_catalina.log-1322240461.gz, etc.
/var/log/tomcat6/catalina.out /var/log/tomcat6/monitor_catalina.log /var/log/tomcat6/tail_catalina.log {
size 1M
missingok
rotate 2
compress
notifempty
copytruncate
dateext
dateformat -%s
lastaction
/bin/chown tomcat:elasticbeanstalk /var/log/tomcat6/*gz; /bin/chmod 664 /var/log/tomcat6/*gz
endscript
}