How do actors stop breathing when playing "dead"?

You hyperventilate right before the take, the same technique as for swimming underwater [not highly recommended underwater, but in air, of course, you can change your mind and breathe again at any time]. Also, bear in mind, you're "dead" so not doing much in the way of running around getting out of breath.
Zombies would be a whole different question and answer ;)

The idea is that you reduce the carbon dioxide levels and slightly increase oxygen levels in your lungs, reducing the urgency to breathe again quite so quickly.

It's also helpful if you're in place a good few minutes before the take, so you have time to relax and not have just rushed to set at the last minute, already breathless.

So, as you hear the 1st AD prepping for the take you breathe big and deep for 5 - 10 seconds. Each 1st AD uses a slightly different but predictable series of commands to the crew...

"Quiet, please. For a take"
"Running up"
"Roll please" [to this a dozen others yell "Turning" right across the set]
Then sound says "speed"; you hear clappers announcing the take; camera ops say "set" [which is always the final signal that everything is ready to go].
"We're turning... and.. "
"Action!"

... you over-breathe until you hear the "and" right before action^[1] - then breathe right out.

Corpses do not breathe.
They do not have inflated chests, so you cannot start with an in-breath, you must have breathed right out.

Corpses that breathe get shouted at ;)
"Cut. Reset. We can see the corpse breathing."
Not a good way to use the studio's money.
Corpses that don't breathe on camera get more work than those who do.

In the rehearsal takes you keep your eyes open and watch the cameras, so you know [and can either count in your head or work from any dialogue you can hear] when you need to be mostl still. At other times you can, of course, breathe shallowly until you know you're in frame. If you're not sure and you think it might be significant - ask. Someone can count you through a rehearsal.

The other thing to note, from an audience perspective, is that long 3-minute scene where you were amazed at the corpse not breathing for the whole thing probably took a day to shoot. He probably had to not breathe for no more than 30 seconds in any individual take.

^[1] The "and" before Action is often significant and highly emphasised. Background [supporting artists/extras/vehicles etc] often need to be in motion before the action actually starts to keep a fluid look to the scene. "Go on and" is a common phrase.

Just a late note on

In some movies you can tell that the picture is paused when it is on the dead character.

I don't think I've ever seen this done [I'm not saying it's never ever been done, but I've never spotted it]. The eye is really very sensitive to movement and a total still image would be quite jarring and noticeable.

After comments and other answers...
Note: it's considerably harder to hold your breath whilst breathed out, as opposed to in. You don't have the lungful of oxygen in reserve and the rise in carbon dioxide is more concentrated.
If you didn't know already, the "urgency to breathe" reflex is not triggered by lack of oxygen, but by increase in carbon dioxide concentration.

A couple more options here.

1. Editing. Let the actor hold their breath for as long as the scene requires, then cut-away for something else, then cut back and miraculously, the actor is still not breathing. Or simply show a still shot while other actors are speaking in the background.

2. Prosthetics. Just use a fake chest shell covering the actor's chest, covering any movements.

   Tony Stark is a fairly extreme example (although most corpses don't have bare chests and hands digging around inside them, making this effect easier):

   

Here is what a professional actor says in an answer to a related question on Quora:

I am an accomplished actor and have had to do scenes where I have died, the first thing to do is to prepare for the shot, the director will tell just how long the camera will be focused on you. Before the shot you will be taken to the makeup department to be made up to look like you are dead if this to be a scene that you are in a coffin at a funeral.

As an example when I did an episode of a murder show on ID Discovery and played a murdered victim where the camera would be focused on me for a period of time, when the director was ready for the shot he would ask for a hand signal when I was ready in which I would take a deep breadth and then hold it and hold my position. The director would the take the shot and film for about 40 to 45 seconds and then cut. In some cases the editor can also cheat a shot using a still image of you in position and your still image is matted into the scene where other actors are moving but you are not.

Another thing is done with some actors when their death involves the eyes to be opened and since you are not really dead your eyes will sometimes blink while some actors can hold this for a few seconds, what’s done then is that the rest of the shot is live but the editor creates a still of only your face and then mattes it back on to your face. There are many things that can be done in post production to accomplish this effect to make it look as real as possible.

There are different ways to perform this scene.

Generally, actors/actresses have practice on breathing. They usually take a deep breath before "action" and leave it very slowly, so you can't see their chest moving.

Besides, the shot is taken in a way that audience don't get time to study their chest movement as you said and sometimes, they only take close up of face not the chest.

If there are still some movements, it gets fixed in post-production or sometimes it just gets ignored.

They are simply holding their breath.

In addition to what was said, with a little bit of training* you can hold your breath 3 minutes. There is hardly a take over 20 seconds in today's movies, so it is very easy.

*Training: during the day, whenever you are bored, like waiting in line, watching TV, or sitting on the bus, just hold your breath. Start with 20 or 30 seconds, and add 5 seconds when it's easy or every day. After three weeks, you'll easily manage 3 to 5 minutes (unless you're chain-smoking or have health issues).

This has been adressed in an answer to this related question on Quora:

In a movie, or single camera TV show, the shot never lingers long enough on the “dead” actor to give away the fact that they are faking it. By cutting away at the right time, the audience doesn’t have enough time to study the “corpse”.

Also, you will see that the DP frames the shot as a medium close up on the actor, instead of holding a wide master which would show the chest movement. That being said, when playing dead, the best technique would be to take a deep breath in before “Action” and then slowly let your air out of your lungs (through the nose) during the shot. This will keep you still and your chest won’t be rising and falling. This is the same technique you use when you swim underwater.

One thing that you cannot control is the pulsating cartorid artery on the side of your neck. That pulse comes from your heartbeat. Sometimes you can see it in close ups of “dead” actors. With CG VFX though, the filmmakers can digitally fix that.

If you’re cast on a show where you are going to play dead, make sure that you avoid the coffee at craft service. The caffeine will make it very hard for you to relax, and your neck will be throbbing away, making it difficult to pull off the shot. Many actors get cast to play murder victims in procedural dramas, so these are some good techniques to keep in mind.