Do soldiers in combat throw their magazines away when they fire all their bullets?

If it's possible at the time to save the magazine they do, but during a running firefight that is not always achievable. 

They do not save the magazines if they are involved in a fire fight.  After it is over they might pick up the magazine but usually they just get new loaded ones from the quartermaster. 

Most times, if in a static battle. You'd detach the mag and stow it. In a running battle. You'd drop that s--- and reload as fast as you could. In 26 months in combat. I never heard of a single Marine getting his a-- chewed for missing mags. Our normal load out was 10 mags with 300 rounds(although we never would load those metal mags with 30 rounds. They tended to have "stovepipe" ejection jams due to the spring tension. 29 was the max). But we'd beg, borrow or steal as many as we could carry.

The answer to your question depends on the nature of the combat.  For example, attacking infantry is moving quickly thru the battle zone.  It might not be practical for the infantryman to recover empty magazines.  In a defensive position, which is very static, it would be extremely easy for a rifleman to retain his magazines.  If the combat (offensive or defense) is extremely intense, it may be about fire-and-load, fire-and-load and the destiny of empty magazines is not important. 

I was in the AF Security Police, now named Security Force, just as Daniel was. We had to fire the AF Qualification Course and a Special qualification course for those armed daily which meant that went to the range twice a year just for the M-16 rifle. Not officialy part of the qualification course but just extra training was the ammo issue. We always received our ammo in cloth bandoliers which had the cartridges in stripper clips in pockets, two stripper clips in each pocket. We became quite proficient in loading the stripper clips into the magazines using the "boot" adaptor. It was generally understood that you had the magazines that you had, you wouldn't get any more and you had to reload during a "lull".

I was trained to stow the magazine. although that wasn't reinforced every time I went to the range.  You can develop the muscle memory to stash them in an ammo pouch or cargo pocket.  If you happen to blow through your 200+ rounds, your resupply might be in boxes of stripper clips, so you would need those old magazines.  

Before WWI, when we were still allies with Germany, Germans invented the Luger pistol.  It was the best semiautomatic pistol yet (predating the Colt 1911).  They sent the US a couple of sample pistols to test, hoping we'd buy them for our military.  They were made in 45mm.  One was destroyed in testing, the other one is like the ultimate collector's gun today, worth millions of dollars.

But while the pistol proved reliable, the US didn't adopt it. One of the main complaints was that the magazines were too expensive to throw away.

In combat, a soldier will empty a magazine, detach it, and just drop it on the ground as he puts a new magazine in.  Soldiers in a battle have more important things to worry about than returning empties for the deposit.