What would be the effects of an expiration date on currency? Let's say we introduce a monetary system where each piece of money is printed with an expiration date - a year, perhaps.  

I'm imagining something rather different from inflation (although I do not know whether it would actually behave much differently) :
With inflation, the entire money supply gradually devalues at some set rate.
With an expiration date, there is no devaluation for a while, and then a sudden drop. However, at that point (and in fact for quite some while already) there will be new money in print, which will still have the same value the old money would have had a day ago (plus/minus the undoubtedly still existing inflation rates on top of that, but unless that's somehow important in this, let's ignore that for the purpose of this question.)

So the setup would essentially be a government infinitely printing money BUT to prevent hyperinflation, old money is suddenly and permanently lost from the economy. Presumably the printing rate would roughly coincide with the expiration date such that, at any given, about the same amount of money (plus/minus external demand but let's ignore that as well) is available. It may just suddenly change owner.

What will happen, short- and long term? (Obviously short time a lot of things, like savings plans and how banks work would have to significantly change or they would downright fall flat, but after a while, assuming our hypothetical government wouldn't just frightenedly revert the laws after two years of resulting economic failure or something, a new economy would build around these new restrictions. What would that be like - without delving too deeply into pure speculation. Are there useful models adaptable/built for such a situation?)
A bill with an expiration date does not become worthless suddenly. Clearly its value would decrease over time until, just before the expiration date, it was next to worthless. Think about it: how much would you pay for a $1 bill that expires in 5 minutes? Certainly a lot less than you would for one that expires in 6 months.

This constantly changing value would make it hard to use as a medium of exchange. Whenever buying or selling something, one would have to look not just at the face value of the bill, but also at the expiration date, to be able to calculate its real value. People would probably start carrying calculators next to their wallets.

Cattle (which obviously "expire") were once used as currency, so it certainly could work. But that does not mean it is ideal.