You could use the Boolean modifier to alter the base mesh (making 'real' holes rather than just the appearance of holes). This can be achieved directly from the image by way of a Displace modifier - however, this is far from efficient due to the very large amount of subdivision required, so be warned!
Add a plane to the scene and add a Texture set to the image to use for the mask.
Add a Subdivision Surface modifier to the plane. Ensure it's set to 'Simple' (so it subdivides without distorting the original mesh). You may need to add multiple Subdivision Surface modifiers to get the required subdivision levels (there's an upper limit of how many levels of subdivision each modifier can apply). Below the Subdivision Surface modifiers, add a Displace modifier using the texture created from your image as shown below :
Adjust the Displace settings (Midlevel and Strength) until you get a surface suitable for carving out from your mesh :
Next you need to ensure your target mesh is manifold - effectively a complete solid - since the Boolean can produce unexpected results if the mesh is not manifold (it cannot determine what is 'in' and what is 'out'). The default Suzanne mesh is not manifold, due to the eyes, and so needs some minor tweaks - I removed the 'eyes' and filled in the sockets to make it manifold :
Now you can add a Boolean modifier. Set the Operation to Intersect and the Solver to 'Carve' (this allows the 'cutting' object to be non-manifold, to slice out sections of the target mesh).
This can produce the following result. Note that you'll need to crank up the levels of subdivision to get sufficient detail from the image for well defined rounded holes (otherwise it becomes very 'pixelated'). The resolution of your image is also significant here.
Note that this method is very CPU intensive due to the very large number of generated faces required. A better method would be to dispose of using an image and to model the 'mask' mesh directly - eg, by creating a cylinder and using Array modifiers to duplicate it into a grid.