Should filling up the tank to full capacity be avoided if possible for the fact that it increases the weight of the vehicle which could reduce the fuel efficiency?
No. Vehicle weight is only one factor of fuel economy. The most important factor is air resistance. Vehicle weight comes as the second most important factor.
My ~1700 kg (car + driver) Toyota RAV4 hybrid has 56 litre fuel tank. Gasoline has a density of 0.735 kg/litre. This is 41 kg of fuel. If you reduce it by half, you save about 20 kg, or a bit over 1% of the car's weight. Taking into account the fact that vehicle weight is only the second most important factor in fuel efficiency, this translates to below 0.5% fuel economy gains.
Your driving style matters far more than the below 0.5% saved by a partially filled fuel tank.
Also, in typical driving, you actually have some of the time the tank half full. If you fill at 25% to 100%, it means the average is 62.5%. You could perhaps fill at 15% to 35%, and now the average would be 25%, meaning the savings would be 37.5%, less than the 50% I assumed previously. So, the 0.5% figure actually was unrealistically high!
Additionally, the sum of the gains you save will be lost by more frequent trips to the gasoline station.
There's actually more going on here than meets the eye, and it all comes down to what kind of driving you are doing and over what terrain. Generally speaking, the best efficiency comes when rolling over perfectly flat terrain, with perfectly steady air (hurricane force tail wind is preferred, but unlikely), at an optimal speed (based on car aerodynamics and transmission/engine design points) and using cruise control (because humans are not as good at keeping the speed fixed). So, to maximize efficiency, you want to minimize accelerations.
So, how does that come into play? Well, mass * acceleration = sum of all forces. Since our car doesn't change shape (hopefully!) when fully loaded with gas, our aerodynamic drag doesn't change. And if we're at a cruise (constant speed), the force the engine sends through the tires offsets the drag. (This is assuming that aerodynamic drag is much more significant than friction drag -- aerodynamics won't change with weight, but friction will -- this is a pretty good approximation). If we're cruising at a perfectly constant speed, we get something like:
force_engine - force_drag = mass * zero (no acceleration)
So at a perfect cruise, mass doesn't matter. But, there's changes in wind, and terrain, and our foot varies the gas and so on. So the forces are always slightly out of balance. If the forces are always slightly out of balance, there will be some accelerations. But how large the accelerations are depends on the mass -- more mass, smaller changes in speed.
Smaller changes in speed means it takes smaller changes in forces to restore the balance and it also means (assuming when cruising the engine is at its optimal efficiency) the engine gets to stay very close to its optimal operating conditions while restoring balance.
Now, the differences in mass between fully loaded with gas and not fully loaded are very small as pointed out in the answer from juhist. The accelerations are a bit less than 1% smaller with a full gas tank compared to half a gas tank. On the other hand, if your car is completely loaded down with things (like mine was when I was moving and had my car loaded with books), the mass difference can be much larger and efficiency is impacted.
Because of the balance of the forces, if you want to be accelerating -- driving in traffic, in the city, stop and go -- then extra mass will kill your efficiency. But, if your only accelerations are on the on-ramp to the highway and then you cruise at nearly constant speed until you need gas again, the extra mass really helps the efficiency. My usual, unloaded car on the highway at cruise gets around 23 miles per gallon. But fully loaded, I can repeatedly get 30 miles per gallon on the same highways.
Weighing down things helps steady-state cruising, stripping out as much weight as possible helps accelerating/stop-and-go driving. But, gas tanks in usual cars don't contribute enough weight to make a significant impact either way.
On an airplane, it's a factor (as is 'tankering' from cheaper airports). It's worse than useless on a car.
Why? Take a freeway cruise where you're only doing half fills. That means twice as often, you exit the freeway and drive the 1/4 mile to the gas station, and that deviation from cruise takes fuel. It wipes out the tiny savings from making the car lighter.
Remember, doing half fills means you have an average 1/4 tank. Doing full fuels means you have an average 1/2 tank. So we're only talking 1/4 tank difference. When you apply that to Juhist's logic above, you can see the most you hope to save is 1/4 of 1%.
Really, this is about trading other resources for increased fuel economy. And another resource we need to talk about is time.
Realistically, having timed it, it is difficult to do a refueling in less than 10 minutes all-inclusive. Let's look at what else we can do with 10 minutes. Half-tanking means adding an extra 10 minutes to a normal tankful. Assuming you spend 3 hours (180min) at freeway speed, what happens if you slow your speed so you spend 3:10 (190min) traveling that same distance?, e.g. From 70 to 66? Your MPG goes up by, in my experience, 3-5%... or more. That's massively larger than 1/4 of 1%!
So there are other ways to save that fuel that work better, all things considered.