What can we deduct as a part of the PilotsNPaws Org, Inc program flying animals to/from their needed destinations?

When you spend money to further the purpose of a qualified charity, you are able to deduct certain out-of-pocket expenses. 

You can deduct what you might consider "hard costs" like the cost of fuel, oil, reasonable maintenance, etc. That is, expenses you would not have made had you not made this trip. 

Note that you may find "cost per mile" numbers for your aircraft. Cost per mile probably includes routine maintenance and repairs. You can claim the cost per mile plus fuel (if the cost per mile does not include that) for the trip. Be sure to document where you got the cost per mile number in case someone asks.

Note that you cannot deduct the value of your time, nor can you deduct the full expenses of a trip that you also took for personal reasons, like flying to see family and carrying a pet along the way. 

But if you are flying a pet and this trip just so happens to give you enough hours for something (I am not a pilot - do you have to fly X number of hours in a certain timeframe to keep your license?), then that would be fine.