Would hydrogen fuel tanks on an airplane increase lift?
Let's do some quick math. Air has a density of about 1.3 g/L and hydrogen has a density of about 0.09 g/L. The best case for lifting force would be storing the hydrogen at the same pressure as the surrounding air. This means that every L of hydrogen can displace 1.21 g of air, reducing the density of the aircraft. The amount of air displaced by the aircraft (and therefore the buoyancy) would not change, but the density of the aircraft in that space would decrease. This would decrease the lift needed, but let's see by how much.

The greatest amount of free volume in an aircraft (that isn't part of the pressurized section of the fuselage) is the fuel tanks. An A380 has a fuel capacity of about 323,000 L. If we consider filling half of that with hydrogen, this would result in a total reduction of about 190 kg. An A380 has a max takeoff weight of 575,000 kg. This change would help but would be very small compared to the total lift needed.

As David Richerby points out, hydrogen needed as fuel would need to be pressurized (if not liquified) to fit enough onboard. This would increase the density above that of any displaced air, which would instead add weight to the aircraft.

Perhaps the hydrogen could be produced instead of stored. Newer aircraft do have something called an OBIGGS, or onboard inert gas generating system. This system generates nitrogen which is used to fill the space in fuel tanks after the fuel is used. Being an inert gas, nitrogen can lower the amount of oxygen in the space below what would be required for ignition of the fuel vapors.

However, this system splits nitrogen out of air, considering that our atmosphere is mostly nitrogen. Hydrogen would have to come from somewhere else, usually water, which would add more weight. Another problem with hydrogen is that when added to air it can be explosive all by itself, which would make the situation even worse. Considering the minor benefit, this would not be a system worth installing on an airliner.

Regarding the feasibility of a hydrogen powered aircraft, NASA did a study and decided that this is not a good source of power for an aircraft. The volume and weight required to store even liquid hydrogen would be too much for the aircraft to be very useful.