UK has four levels of departure taxes for departures from UK, which includes stopovers. The higher level applies for any cabin that is not the lowest on the a/c (ie premium economy, business and first). This applies to all airlines, not just BA.

Some other countries have luxury taxes for business and first class tickets purchased from that country (eg India) which need to be factored into your purchase decision.

Most other taxes do not vary by class of travel, however airport taxes vary by airport from almost nothing to quite sizeable amounts (eg SYD).

Fuel surcharges vary considerably by airline. It is the ticketing airline that decides how much fuel surcharge to collect - some have few charges (eg AA) whilst others have very high surcharges (eg QF).