Clear Heroku logs (command)
Currently, there is no way to clear the Heroku logs. Heroku maintains a record of the past 1500 lines of log activity.
heroku restart
will clear logs - although if you're running in production you may not want to do this on a whim :D
Other than that you can use the advanced logging addon (Syslog drains), which gives you the ability to clear the logs:
deprecated
heroku logs:drains clear
Update 2019/08/07
heroku drains
More info: http://devcenter.heroku.com/articles/logging
The current command to clear logs is
heroku drains