All aircraft are subject to performance changes due to fuel burn, and since weight, lift, thrust and drag are all tied together for every aircraft in the same way the changes scale nicely.
Let's look at some extremes: One of the most interesting examples of this was the Concorde.
Since the Concorde flew ~20K above every one else in the sky in those days it didn't have the same altitude holding restrictions that regular airliners (or any IFR airplane) has.
The Concorde was give a block altitude basically to do what it wanted up around 50,000 feet (ca.
15 km).
Since it burned quite a bit of fuel at Mach 2.1 and gradually became lighter during the flight the pilots let the plane slowly drift up to gain efficiency as well as gliding distance if needed.
The Concorde also used its many fuel tanks to trim the aircraft as well as cool the nose cone.
Fuel burn was closely monitored and constantly pumped around to ensure the aircraft remained in trim.
Even the dinky Piper Archer
I fly is affected by partial tanks.
If I fly the tanks from full to VFR legal reserves, the CG moves forward a full inch.
Granted this does not seem like a lot you will need to trim to compensate for this throughout the flight.
You will also be landing a full 240 LBS lighter which is about 10% of the total weight from takeoff.
If you come in hot and high you can get yourself into trouble.
Yes you can feel it, generally pilots are not going to do “faster” maneuvers but the maximum G load may change ( see your aircraft's POH for more info ).
Any smart pilot should practice flying their aircraft at various fuel loads to understand the differences it makes in performance and handling.
Some aircraft with things like tip tanks or auxiliary tanks even have fuel burn limitations that define where the fuel is to be burned from and in what order, again going from something as small as a Bonanza all the way up to the Concorde as discussed above.
Just how much can the handling change?
Assuming you did your due diligence in pre-flight and calculated your W&B on both ends of the flight accounting for fuel burn to ensure you are within your CG limits
the flight should occur with in the limits of the aircraft's POH and generally speaking within operable limits of the airframe.
FAA certified aircraft have control force limits so even at the extents of the CG range
you should be able to trim the aircraft to handle “normally”.
You will likely notice pitch differences (at either extreme) as well as various performance number differences.
aviation.stackexchange.com/users/3677/Dave
Modern gliders are an interesting data point regarding this.
A competition glider carries water in its wing tanks to increase weight.
This betters performance at high speed, which is essential for competition when conditions provide strong lift.
The pilot can release the water at will, and the tanks can be emptied in a few minutes.
This happens in competition every day, either when you are arriving (to land without the extra weight) or sometimes mid-flight when conditions are worse than expected.
So, taking as an example a Ventus 3: This glider, with pilot included, weighs around 380kg.
It can carry up to 212 litres of water in its wings, for a maximum gross weight of 600kg (it also carries a few litres in the tail for cg purposes).
In summary, the pilot can change the weight of the glider from 600kg to 380 kg in some 4 minutes (meaning a 38% change in weight!).
This makes the change very noticeable.
When empty, the glider is more "feisty", both in that it reacts more to turbulence, and in that the commands (the ailerons, especially) are way more effective.
aviation.stackexchange.com/users/690/Martin Argerami
Aircraft performance does change with weight.
Not only does takeoff weight and balance need to be considered, zero-fuel weight and balance does as well.
Many key performance metrics depend on this, like takeoff and landing speeds and distances.
Climb performance and range are also affected.
Inflight, Va (maneuvering speed) is very dependent on weight.
The speed at which you can fly while putting extreme loads on the airframe from abrupt maneuvers or turbulence actually decreases as weight decreases.
aviation.stackexchange.com/users/46356/Dean F.
Something else that changes with a change in CG and/or weight is your stall speed.
There is a good article from AOPA here: https://www.aopa.org/news-and-media/all-news/2020/march/flight-training-magazine/aerodynamics-stall-speed
aviation.stackexchange.com/users/50060/MikeG
Alone in an A36 Bonanza from central Florida to North Central Indiana, I flew on one load of fuel by decreasing the throttle gradually as I burned fuel to maintain a constant airspeed.
The aircraft wanted to speed up as it lost weight.
So, yes the aircraft will fly differently as the fuel is used.
aviation.stackexchange.com/users/26618/Mike Brass
Modern aircraft are capable of carrying a lot of fuel to achieve extra-long range; we're at a point where nearly every pair of destinations can be reached non-stop. Carrying so much fuel imposes limits on take-off and landing weight, so the proportion of fuel to airplane weight is not negligible.  I'm wondering how the handling characteristics of airplanes change as the fuel is used up. Do the pilots feel that the airplane is lighter towards the end of the flight? Can they do faster maneuvers with nearly empty fuel tanks?
How do handling characteristics depend on flying with a full vs. an empty tank?
aviation.stackexchange.com
<fuel><weight-and-balance><flight-dynamics>
aviation.stackexchange.com/questions/78869
aviation.stackexchange.com/users/3291/JohnEye
Aircraft performance does change with weight. Pitch differences (at either extreme) are noticeable as well as various performance number differences. All aircraft are subject to performance changes due to fuel burn,  and since weight, lift, thrust and drag are all tied together for every aircraft  in the same way the changes scale nicely. Many key performance metrics depend on this, like takeoff and landing speeds and distances. Moreover, climb performance and range are also affected. As a general rule, aircraft performance can be affected by weight, including fuel weight. Examples illustrate that, as aircrafts consume fuel, they become lighter, gain efficiency and display pitch differences in various values. Gliders, on the other hand, are designed to carry water for extra weight. This design improves performance at certain speeds, as it makes a stronger lift possible, facilitates aileron operation and stabilizes the glider body. Areas affected by weight change include takeoff and landing, climbing and in-flight. On the same note, it is worth noting that when manoeuvres add load to the airframe, turbulence tends to decrease along with weight, even though aircrafts generally increase their speed when losing weight. 
As a general rule, aircraft performance can be affected by weight, including fuel weight. Examples illustrate that, as aircrafts consume fuel, they become lighter, gain efficiency and display pitch differences in various values. Gliders, on the other hand, are designed to carry water for extra weight. This design improves performance at certain speeds, as it makes a stronger lift possible, facilitates aileron operation and stabilizes the glider body. Areas affected by weight change include takeoff and landing, climbing and in-flight.  It is worth noting that when manoeuvres add load to the airframe, turbulence tends to decrease along with weight, even though aircrafts generally increase their speed when losing weight.