Yes. In the case of your paper, you are supplying a jet of air. By doing so, the static pressure of that 'streamline' of air must be lower (due to Bernoulli). The paper then feels the low pressure of air flowing over top of it, and rises accordingly. In an airfoil, it does not work the same way because you are not supplying a jet of air. The airfoil is moving through the air and reducing the pressure* due to it's shape.
Well, keep in mind that Bernoulli's equation is valid along a streamline for steady flow and we assume viscous effects are negligible. None of this is true if you have a flapping airfoil, so it is incorrect to try and apply this equation to your oscillating airfoil.
*In the first 1/3, the airfoil reduces the pressure. But on the last 2/3rds it increases the pressure back to static at the trailing edge.