Would an amalgam filling under or next to a gold crown cause any electrochemical reaction?
The answer is basically yes, if the dental procedures are done improperly.  Here is a good answer from a DDS:


  Galvanic current is a term that has been used in dentistry for over
  100 years. It is a condition created by the presence of dissimilar
  metals in the oral cavity of the teeth and gums, with saliva serving
  as the electrolyte. Have you ever felt a “shock” to your teeth caused
  by a piece of tin foil or a spoon that touches a sliver or mercury
  filling in your mouth? If you answer yes to this question, you’ve
  experienced a galvanic event.
  
  There are several different types of galvanism: 1) A silver/mercury
  filling is placed in opposition or adjacent to a tooth restored with
  gold. These dissimilar metals in conjunction with saliva and body
  fluids constitute an electric cell. When brought into contact, the
  circuit is shorted, the flow of electrical current passes through the
  pulp, and the patient experiences pain. 2) Dissimilar metals coming
  into contact when the upper and lower teeth come together and touch
  each other. 3) Two adjacent teeth are restored with dissimilar metals.
  The current flows from metal to metal through the dentine, bone and
  tissue fluids of both teeth resulting in discomfort and tooth
  sensitivity.


Additionally, from the same site:  


  Metals will not necessarily always cause the galvanic effect in the
  mouth. It depends on the specific metal and alloy being used along
  with quantity and placement in the mouth.


In other words the amalgam filling and the gold cap may form a weak galvanic cell.  The proximity of the amalgam to tooth nerves creates a situation in which only a very weak cell current is required in order for the resulting sensation be perceived.  The rate of degradation could thus be very small, or if the filling and capping procedures are done correctly there will be no galvanic affect at all.