No, it doesn't.
The session will continue until the tutor and/or the student decides to quit.
If you have 2 sessions back to back with the same student, you should stay with the for 1 full hour (no need to quit and reconnect). You must, however, submit 2 feedback forms for each of the sessions taught (you should copy and paste the content from one form to another, set the session goal in both of them, as well as evaluate your student's speaking performance).
If you have 2 sessions booked back to back but with 2 different students, you should look at the time and start saying your goodbyes at 29:00 so that you can finish off in time to connect to your next student. It's ok if you are a few seconds late but be careful about time-keeping.
In order to quickly move from one student to the next, you can simply refresh the page at the start of your second session, and it you will get connected to the new student.