Are any special rules added if you are playing with more than one expansion?
Short answer: No. There are no special rules for playing with more expansion.

Additional info: However, there were some special rules for playing with specific expansions.

On January 12 2016, FFG had posted an update which featured new Reference cards with a following description:


  Due to the introduction of new mechanics in its expansions, the balance of Eldritch Horror has shifted. To remedy this, revised 
  Reference cards have been created for use when playing with any of the following expansions: Mountains of Madness, Strange 
  Remnants, Under the Pyramids. Below are the revised Reference cards.


This update only lasted a week. On January 19 another update renamed this cards to "variant" not "revised" reference cards.


  With the introduction of new mechanics in Eldritch Horror expansions, some player groups may wish for a more difficult or more
  relaxed experience with certain player counts. The variant Reference cards shown below can be used to increase or decrease the
  game’s difficulty.


I think that the revised reference cards are what you are asking about.
Arkham Horror was becoming harder with each expansion added and needed a special rules to fix that.
Eldritch Horror was becoming a bit easier with each expansion added (other than "Forsaken Lore") and these cards have returned the difficulty back where it was in the core game.
That's my experience, at least.
You may find these new reference cards at the end of the official FAQ.