No. Your course will remain as long as you have an active account (paid or free).
There is no deadline for you to take a course but a few things to note:
1. The average completion time is 7 days and
2. After 14 days you are much much much much less likely to complete the course
So, while you have as much time as you like to complete, we encourage you to a) start sooner rather than later and/or b) once you start, try to complete within a few weeks.