When you reach the limit, Bugfender will stop logging any new sessions (a session is an execution of the application on a specific mobile device). That means currently running sessions will be still logged in full, but new sessions will not be logged at all. Bugfender will slow down the acceptance from that point on, to match the amount of logs permitted by your plan.
Of course, you can upgrade to lift your log limit immediately!
Most importantly, Bugfender will be just disabled but your application will run normally.