In the paper clip experiment, the surface tension of water prevents the clip from falling, thus we can assume it exerts a force of $mg$ (weight of clip) upwards.
However, if you try to pull the clip out of water, surface tension opposes this motion also. If for simplicity we assume that the clip is a rectangle of length $l$ and breadth $b$, and the surface tension is $T$, then the liquid exerts a downward force of: $$T×2(l+b)$$
What is going on here? Why is there an inconsistency in the direction along which the force acts?