Is the Pope allowed to publish books after resignation? I've heard that after popes die (at least after Bl. John Paul II died) all their unpublished works and letters are burnt and I heard on the radio this morning that Pope Benedict XVI is pretty much going in to hiding, not just as a former head of state, but also to not influence in any way the new Pope. 

So, we were expecting an encyclical on Faith for the year of Faith 2013, and if it doesn't get published in the next two and a half weeks will it be destroyed or can it (if it even exists) be published "posthumously"?
The ex-pope can't publish anything as a pope - his seal is destroyed and he has no longer the right to use his papal name in this way. Documents he didn't manage to publish won't be destroyed, but are open to use by his successor, as the site linked in Svidgen's comment states.

No one can forbid the ex-pope to publish books under his civic name, though it's a tradition for an ex-pope to refrain from anything that might have "political" consequences. I personally (with no qualification, I just guess) expect Joseph Ratzinger to publish some of his yet unpublished works (otherwise, someone would publish them posthumously) - I guess he has enough unpublished (or unfinished) materials from his academic times.