What biological processes can affect the temperature of the shadow a tree casts?
The short answer is NO. A shadow is strictly due to the blocking of sunlight, so the shadow of 2 equally non-transparent (i.e., same density of leaves, both have similar trunk diameter, etc.) trees on the same hillside in the same weather conditions would not have different temperature shadows.

A plant's metabolism can and will affect the temperature surrounding the tree, but this affect is lost almost immediately as you move away from a tree. For example, in a study I did a few years ago measuring spruce tree temperatures (unpublished), we found that even within 5-10cm from the tree, the rise in ambient temperature was no longer measurable. 

You're correct that the leaves of plants will absorb sunlight and as a result will increase in temperature. Walter Loomis's (1965) paper, Absorption of radiant energy by leaves (found here), suggests that leaf temperatures can raise to 30 $^\circ$C or more. However, he notes that their study found that even wind speeds of 5mph instantly cooled the leaves via conductive heat transfer. Our study with spruce trees also found this to be true. Transpiration will further help the tree rid itself of that increased temperature fairly quickly. Regardless of these facts, radiant energy from leaves in the canopy will not be felt how-ever many meters below within its shadow. 

Finally, yes, transpiration can affect local climate, but this is only on the scale of whole forests. The transpiration of a single tree will likely not impact the ambient temperature around itself in any measurable way, especially at any distance greater than that I've already mentioned.