Do aircraft weigh less on an eastward flight than a westward flight, due to the Eötvös effect?
Technically yes. But as Jeffrey said, it's quite negligible. We can do the math knowing that the earth rotational speed is $360°$ per $24h$, you get $0.004°$ per second or $\omega =  7.27 \cdot 10^{-5} rad/s$. We can then calculate the centrifugal acceleration related to this speed assuming the earth radius is $r=6371km$ and neglecting flight altitude.
$$a_c = \omega^2 r= 0.0337m/s^2$$
So if you're flying at a relative ground speed equal to the earth rotation, your centrifugal acceleration is either $0$ or $0.134m/s^2$ depending on which direction you are flying. Compare that to the gravitational acceleration $g=9.81m/s^2$ and you get $1.37\%$ of weight difference flying at Mach 1.5 with the earth rotation or against it.
Now remember that to have your weight reduced you need to fly with the earth rotation which means eastwards. And there you get a somewhat bigger problem reducing your performances in flight. Jet stream. Jet stream magnitude can be up to several hundreds of kilometer per hour meaning reduced ground speed around 10% are quite usual, resulting in 10% increase of flight time and fuel consumption thus destroying any opportunity to notice the 1.37% reduction of lift you're getting by flying east.
And even less useful when you know that the concorde used 2 tons of fuel to taxi from gate to Runway and 50 more tons to reach cruise speed and altitude in less than 45min....