A. Renaissance paintings were created in conditions involving far greater fluctuations in temperature and humidity than those permitted by current standards. --> This strengthens the argument but not an assumption in anyway.

B. Under the current standards that museums use when storing Renaissance oil paintings, those paintings do not deteriorate at all. --> again strengthens but not an assumption
C. Museum collections typically do not contain items that are more likely to be vulnerable to fluctuations in temperature and humidity than Renaissance oil paintings. --> It is clearly stated that such methods are used only when 'storing Renaissance oil paintings'
D. None of the materials in Renaissance oil paintings other than the paint are vulnerable enough to relatively wide fluctuations in temperature and humidity to cause damage to the paintings. --> the argument mentions 'old paint' as being resistant, clearly assuming the canvas painted on and maybe even the frame are also resistant.
E. Most Renaissance oil paintings are stored in museums located in regions near the regions where the paintings were created. --> No mention of any regions.

So D.