Can membership level drop due to inactivity?
No. You just lose activity due to inactivity, so you could miss out on being upgraded faster.
You don't lose activity if you are inactive, but if you don't post in any given 14-day period then it isn't added to the "time" value of the activity equation.
The activity number is determined in this way:
time = number of two-week periods in which you've posted since your registration
activity = min(time * 14, posts)
Activity is updated every hour.
This thread has more information on the specifics of activity calculation and forum rank requirements:
https://bitcointalk.org/index.php?topic=178608.0