Technically, the sonic screwdriver shouldn't be able to do nearly as much as it does in the series (open doors, reprogram Cybermen, adjust any of several other devices of largely arbitrary function). It is, sad to say, a magical "do anything" device.
As such, throwing in an arbitrary weakness, like being unable to do anything to any wooden objects, allows the writers to add tension to a story. If your all-powerful device suddenly can't do anything, then you are in a bit of trouble and need to find a new way out.
I honestly don't know if there is an explanation that involves the in-universe "science", but the inability of the sonic screwdriver to work with wood seems to be an addition to keep it from being used in too many deus ex machina plot resolutions.