What physical/chemical reasons could have caused Dorian Gray's picture to degrade?
Preface: So far, I do not know the novel you refer to, and assume the portrait is a classical oil painting.

Such a painting consists of several layers of products applied.  Similar to a wall in your living room, the canvas is prepared to accomodate the pigments dispersed in linseed oil, what in other fields of application would be called a "primer".  Then there are multiple layers of the painting itself, topped by the varnish.  To mention a few of the potential issues, 


Pigments may change their colour upon exposure to air. Pigments like lead white turned over time black, for example.
Pigments may interact with the oil, forming complexes, of different mechanical and complexation strength: 




(source)


Components may migrate across the picture and locally, and reactions may occur.  The formation of soaps of lead and zinc are the most prominently known ones, for example based on palmitic, stearic, or azelaic acid.  These processes may alter colors, even rendering them transparent enough to recognise the support of the paint, and they may eventually protrude the surface of the paintings as shown from a topview:




(source, p. 14)

or cross section:



(source)

These may not simply be "washed off" as a normal cleansing would do, as water and humidity in general amplify the problem.  If left untreated, the painting however will age and degrade.

Further reading may be found, for example, here, here, here, here/here.