Do airlines in the US or UK have to pay fuel taxes?
In the US, yes.

Kerosene [for use in aviation] is taxed at $0.244 per gallon unless a reduced rate applies.
These taxes mainly fund airport and Air Traffic Control operations by the Federal Aviation Administration (FAA), of which commercial aviation is the biggest user (Fuel taxes in the United States).

More information from taxfoundation.org.

In the UK, it depends on the user.

Aviation turbine fuel used for private pleasure flying: £0.5795 per litre
Aviation gasoline (Avgas): £0.3770 per litre

Air transport is exempt.
Furthermore:

Article 24 of the [1944 Convention of International Civil Aviation] requires all contracting states not to charge duty on aviation fuel already on board any aircraft that has arrived in their territory from another contracting state. Further to this, the exemption of airlines from national taxes and customs duties on a range of aviation-related goods, including parts, stores and fuel is a standard element of the network of bilateral ‘Air Service Agreements’ (ASAs) between individual countries.