No. Wood does not have any magnetic response at DC (i.e., at low frequencies like when using a bar magnet or a refrigerator magnet to move around metal shavings). Of course if the ruler is too thick then the shavings will be too far away from the magnet and nothing will happen, but then it is not the wood's fault, it is only because of the distance between the magnet and the metal shavings.
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Yes it does. All things can magnetize but the molecules of wood are complex enough so that the vector sum of the individual magnetic dipoles is close to zero with a lot of effort needed to get the wood to "magnetize". What I'm saying is that the average magnet you would come across is the result of the orbital velocities of electrons around an atom, so if the orbital momentum's of each atom are parallel you will feel a "magnetic force". (I know quantum mechanics people that the electron orbits are not circular, but this is a standard enough way to look at it to allow intuition into the problem).