Is energy expended when a force is exerted on a object? Energy is expended when a person exerts force on a object. 
1) From the equation Work Done = Force x Distance, if a force is exerted, but no distance moved, then no work is done. Hence, the statement is false?
2) However, when a person exerts a force, isn't  he expending chemical potential energy? In other words, there is an energy change from chemical to thermal. Hence, the statement is true?
I am not sure which of the above is right. 
 A: 1.) As you assumed, since the object does not move any distance, then by the definition of work, no work is done and thus the statement is true - no matter how much force is exerted, if there is no displacement, no work is performed.
2.) If a person exerts a large force on an object and the object does not move, there is no external work done. The muscles in the person applying the force are contracting isometrically yet no external work is performed by the shortening or lengthening of the muscle fibers. However, a person will be fatigued which implies that there is in fact an energy expenditure. In muscles, this potential energy is stored in the series elastic component of the muscle. Internal work is performed by the contraction and releasing of the muscle fibers much like a spring. Again using the definition of work, we can say that this statement is false.
In physics, work is generally referred to external work. 
A: The first statement is an idealisation when no environmental effects are taken into account (nor heat). 
The second is more realistic and takes into account environmental effects in a thermodynamic setting. 
So for an idealisation it would be true while in a thermodynamic setting it would be false.
Now, one basic issue here is the definition of Work, which is not equal to any energy exchange in a system but only to specific energy changes related to given applied force (and only to that).
A: 
1) From the equation Work Done = Force x Distance, if a force is
  exerted, but no distance moved, then no work is done. Hence, the
  statement is false?
2) However, when a person exerts a force, is he not expending chemical
  potential energy? In other words, there is an energy change from
  chemical to thermal. Hence, the statement is true?
I am not sure which of the above is right

They are both right, even if this is paradoxical. If a force is applied to an object and this does not move no 'mechanical work' is done according to newtonian physics even though energy is spent. This happens because that energy spent is turned into other form of energy but does not become kinetic energy and there is no dislocation.
If you want to understand the historical reasons why this peculiar definition was chosen, you can find detailed explanations a this links : work follow the links there, and wasted energy
A: Not necessary.
Example: normal force of physics textbook resting on desk.
