For a setup like the below image, would there be a lift generated to the tube, due to bernoulli effect?
No, there won't be lift. The Bernouilli effect will draw some air through the hole until the increased air density balances out the Bernouilli effect and the pressure is the same inside and outside. When the fan stops the opposite will happen, the air inside, now at higher pressure, will escape through the hole until equilibrium is reached again.
In both cases there will be a force due to the pressure imbalance, but it will not be sustained: it will only last as long as it takes to balance the pressures again. Moreover, due to the small size of the hole, this force will be rather small.
You'd obtain the same effect by pumping some air out of the tube before opening the hole at the top: as you can imagine, this is unlikely to lift the tube at all, and if it does it will not be for long.
To get a lift you have to think of the balance of forces. Lift means against the direction of gravity. If air is blown out against the direction of gravity there will be a lift . The location of the hole will be important, i.e. at the bottom in the spot where there is strong air flow, and the leak will be against gravity.
I'm pretty sure it won't generate much lift. Unless there's something I missed.
You might get a small lift from Bernoulli force. Air rushing past the small hole will have reduced pressure, so air will rush in from outside. Air down means everything else up. Soon the air pressure inside will rise to the point that it balances the Bernoulli force. At that time the air inside will weight slightly more than the same volume of air outside, so the whole thing will be slightly heavier.
I can't see anything else affecting lift. What happens inside the cavity ought to mostly stay inside.