To lift an object, do we need a force equal to its weight, or greater than its weight?

There are two points to be clarified here.

• The normal reaction force from the surface is a self-adjusting force. In particular, it can take any value so as to prevent the object in contact from penetrating. So, if an object resting on a surface has a weight $w$ then the normal reaction force would be $w$ in the upward direction. Now, if you apply an external upward force on the object (with your hand, say) of a magnitude $w/2$ then the normal reaction force from the surface would change its value to $w/2$. Now, if you apply an external force of a magnitude $w$ in the upward direction then the normal reaction force from the surface would reduce to zero.
• However, as you correctly notice, when the upward external force is exactly the same as the weight in magnitude, the object is still in perfect equilibrium. And since the initial velocity of it was zero, its velocity would still remain zero because equilibrium means no acceleration. So, there would be no movement. So, in order to actually lift the object, you do need to provide an upward force which is at least slightly greater than the weight of the object. Once you apply such a force even for a tiny amount of time, the object would pick up an upward velocity because it would have been subjected to an upward acceleration for that tiny amount of time. Once this is accomplished, you can reduce the magnitude of the upward force to be exactly the same as the magnitude of weight and the object will continue to move in the upward direction, in equilibrium, but now, with a constant velocity (that it picked up during that tiny amount of time of acceleration).

In real situations, when a mass is sitting on a table, its weight will slightly flex the table. So lifting with a force that equals the weight will remove normal force pressure from the surface, the surface will flex back to it's former position giving the slight increase in net upward force to start acceleration. This can be visualized easier if you imagine the mass resting on a spring. So in most real situations applying a force equal to it's weight would lift it.

You are correct that there must initially be a net upward force, no matter how small and how brief, to get the object going. But in addition to balancing the forces immediately after to achieve constant velocity there must be a net downward force just before reaching the height such that the object will come to rest, if it is to possess ONLY gravitational potential energy at that height. Otherwise the object will possess both gravitational potential energy and kinetic energy when it reaches that height.

Hope this helps