heroku restart will clear logs – although if you’re running in production you may not want to do this on a whim 😀
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