Why is it possible to use lipids to recover diamonds in mining?
According to the Wikipedia piece on diamonds:


  Diamonds are naturally lipophilic and hydrophobic, which means the diamonds' surface cannot be wet by water but can be easily wet and stuck by oil. This property can be utilized to extract diamonds using oil when making synthetic diamonds.


It looks like the surface of diamond has dangling bonds terminated with hydrogen, so your non-polar hypothesis is correct, as applied to terminal moieties on the surface. See this article about the material properties of diamond where it is stated:


  Diamond surfaces are hydrophobic when the surface carbon atoms terminate with a hydrogen atom and hydrophilic when the surface atoms terminate with an oxygen atom or hydroxyl radical.


and


  Naturally occurring diamonds have a surface with less than a half monolayer coverage of oxygen, the balance being hydrogen and the behavior is moderately hydrophobic. This allows for separation from other minerals at the mine using the so-called "grease-belt."


You ask:


  If so, would diamonds stick to wax or plastic?


I believe the answer is yes, but certainly factors such as the transport of diamond with grit in a slurry over/through a hydrophobic material such as the two you name might come into play: I think mixing the mined slurry with goop is more effective than sheeting it across a 2-D material, and a higher extract yield results.

Note that at higher temperatures and with other reactants, the surface properties can change (as stated in the linked article immediately above):


  Treatment with gases or plasmas containing the appropriate gas, at temperatures of 450 °C or higher, can change the surface property completely.