As far as I can tell, the shaders determine performance, not the textures themselves. The only exception to this is texture size, which can affect things a bit.
To illustrate this, the diffuse shader just uses the color information on each pixel of the texture when rendering it onto an object, as far as I can tell. You can put alpha in there, you can add transparencies in Photoshop, etc…, but it will still just render the pixels and ignore the extras.
If there is a performance difference, it’s way too small for me to measure.