While they are swimming, do sharks move their heads because they have to or because they want to? In other words, can a shark swim without moving its head?

and do dolphins move their head up and down because they want to or because it's just a reaction of them swimming? I have never seen a dolphin (or a shark) swimming (actively) without moving it's head up and down (or side to side).
I've searched google but can't find anything

It has always been my understanding that sharks must keep moving to breath. The head and tail act like a rudder and oars or paddle, they use them to keep moving.
In other words, a shark can swim without moving its head, but not for long. If it does, their tail will tire, they will lose momentum and suffocate.

Sharks have gill slits but no gill-cover, which is common to most bony fish. Because they have no swim bladder to keep them buoyant, sharks sink when not swimming. Shark's bodies are heavier than water and most sharks are constantly on the move. By moving forward with their mouths open, sharks move water across their gills for breathing.
http://www.indianchild.com/sharks.htm

It would make sense that it is like a see-saw action, given the laws of physics re action and reaction etc., that as one end goes down the other end goes up.
Sharks have a vertical tail fin that they move side to side to generate propulsion.
Dolphins have a horizontal tail fin that they move up and down.
Given that when swimming they are in a liquid and are not pushing against a fixed and solid object to generate the propulsion force, it makes sense that the end opposite the tail has to move in a similar (and inverse) manner.

RE BB's answer and keeping on the move.
This used to be the line of thought.
Several types of shark have now been observed "sleeping" and totally stationary on the sea bed for very long periods of time, so this is no longer considered to be the case.

Shark senses
Sharks have two senses that many animals do not have:
Ampullae of Lorenzini
These small pits in the head detect electricity. The shark has the greatest electricity sensitivity known in all animals. This sense is used to find prey hidden in sand in bottom feeding sharks, by detecting the nerve impulses. It is this sense that sometimes confuses a shark into attacking a boat, when the metal interacts with the salt water.
Lateral line
This system is found in most fishes, including sharks. It is used to detect motion or 'sound' in the water. The shark uses this to detect other organisms moving, especially wounded fish. The shark can 'hear' frequencies in the range of 25 to 50 Hz using this sense.

Can they get by without moving their heads? Maybe, but not for long. Do they? No.