I have all kinds of spare cell phone batteries laying around...Can I use them in my Tiger Moth or other slow park flyer? I know I need a special Li-on charger, but if I buy one, will they work? What kind of configuration would you suggest?

Each cell is 3.6 volts nominal.
you will need atleast two cells.
The chargers that come with the phones will charge the number of cells that it is designed for. Some phones have two cells inside the plastic shell and some have one.
Since I do not have a two cell charger, I individually charge each cell and then use an adapter that connects both of them in series.
Most if not all of the modern cell phones are powerd by one cell now.

The protection circuit will not allow over drainage of the battery.
It also may regulate the charging. Much debate on this.
The biggest concern is it shuts down the battery on low voltage and does not rearm on shutting throttle down as the power to everything in the plane is off unless you have a separate receiver battery.