The answer varies according to where you are.
In the UK, you can claim back
duty on the entire content of the fuel tank(s) when going abroad, and it is therefore legitimate for the pump to not charge this duty if they can see that you are departing abroad right away. Not many other countries operate this duty drawback - I have heard that Italy does it but have no idea how one would claim it back... I have certainly never been offered a discount at an Italian pump when leaving Italy.
Claiming back
VAT is something else. AIUI, one is not entitled to do that unless it is a business trip and the aircraft operator is a VAT registered person or business. If the plane is owned by such an entity, the VAT will get claimed back as one does in the normal course of business. So, a typical business flight abroad, on a business owned plane, will pay neither the duty nor the VAT.
It is also possible to get duty free avgas in most of Europe, if one produces an AOC at the pump. I always ask about this and always get this answer. The UK is the only exception I know about, where it doesn't work at the pump but
AFAIK the business (the AOC holder) does a separate reclaim from HMRC.
I would expect the VAT situation to be similar throughout the EU (reclaim possible only on a business trip) but the duty situation is going to be very variable.