Does an increase in lift always cause an increase in lift-induced drag?
When looking at statements like these, it's important to consider which parameters are varied and which parameters remain fixed.

Let's get this out of the way first by correcting your statements:


Lift increases with increase in airspeed with constant angle of attack but induced drag reduces with increase in airspeed with lift force remaining constant
Higher Aspect ratio wings generates greater lift with constant wing surface area but creates less induced drag with lift force remaining constant.


As you can see, each half of your statement refers to a different scenario. The bottom line is that for lift generation, it's more efficient to give a small change of velocity to a large air mass, than a large change of velocity to a small air mass (because force is generated by momentum change which is linear in velocity, and induced drag is due to the kinetic energy imparted on the air mass which is quadratic in velocity).

When flying fast or with slender (high aspect ratio) wings, you affect a larger air mass every second. You can do two things: either keep your angle of attack constant and get a lot of lift out of the air mass, or in case you don't want to do aerobatic maneuvers, you decrease the angle of attack until lift once more equals weight, and get less induced drag.