Does dark energy work like gravity, but the opposite way? If a body has more mass gravity will exert a greater force on it.
Does that apply also to dark energy? In other words, if a body has more mass, will it be affected more by dark energy? (that is, will it be pushed with a greater acceleration from another body)?
 A: Dark energy is a concept devised to help explain the expansion of the universe.  It is presumed to comprise about 68% of the universe (on a mass-equivalence basis), but it is spread so uniformly throughout the universe that its density is on the order of only 10 to the minus 27 kilogram per cubic meter.
Dark energy is not presumed to clump in matter, but rather to exist as a uniform distribution, either as a cosmological constant invariable through time, or as a scalar field that may change with time and/or vary with position.
As dark energy provides negative pressure, rather than attractive pressure, it acts against gravity.  But because its density is sparse and uniform throughout space, it would seem to have no measurable effect on, or in the immediate vicinity of, massive objects.  If dark energy clumped within massive objects, I would expect the apparent expansion of the universe to be less than otherwise, as all objects, including we and our measuring instruments, might be expanding along with or even more rapidly than space.  In that case, space might appear to us to be contracting.
In other words, if dark energy is uniformly distributed throughout space, without regard to the presence of ordinary mass, one can NOT say that the amount of dark energy within the space occupied by a massive object is greater than the amount of dark energy within the space occupied by a less massive object.  
So, to answer your question, NO, not necessarily.
A: As a commenter noted, dark energy, whatever it turns out to be, is not thought of as a property of any known object.
From the Wikipedia article "Dark energy":

In physical cosmology and astronomy, dark energy is an unknown form of
  energy which is thought by some physicists to permeate all of space,
  tending to accelerate the expansion of the universe.

...

Two proposed forms for dark energy are the cosmological constant, a
  constant energy density filling space homogeneously, and scalar
  fields such as quintessence or moduli, dynamic quantities whose energy
  density can vary in time and space.

...

The simplest explanation for dark energy is that it is simply the
  "cost of having space": that is, a volume of space has some intrinsic,
  fundamental energy.

...

In quintessence models of dark energy, the observed acceleration of
  the scale factor is caused by the potential energy of a dynamical
  field, referred to as quintessence field.

A: The answer is no.
Dark energy does not exert a force between objects, but causes space to expand. Objects floating in space are carried along, but remain still in space. That means that if you place two objects in a universe with dark energy, sufficiently far apart that their mutual graviational attraction can be neglected, the speed with which they recede depends only on their distance from each other, not on their masses.
A: We know that is not regular energy nor is regular matter.
Dark energy must clump as per space is distorted from its flat symetry (space is dragged like a blanked would be so more cubic space dragged by a massive object than by one less massive) in places where you have concentrated mass like neutron stars and blackholes. 
There seems to be the posibility that it may clump on very large scales. 
What could be:  


*

*A Quantum field - Big trouble as it does not behave like one as per its energy 
equivalent equal to the mass of 3.7 protons per cubic meter.

*Some portions of relativity need some serious reworking. 

*Einstein Cosmological constant is more comples than thought. 


it has no entropy (and hence no temperature) associated with it and also no dynamics that would support waves. 
At first glance Dark energy is disasociated with space time as per its constant value with expansion and somehow driven to change by another mechanism as per its 10 or so changes that have been detected since the birth of the universe. It seems to interact with gravity as per the effect that exerts on space seen by the acceleration of the universe. 
