It sounds like you are describing two different problems.

FIRST PROBLEM: Turning off a hot water heater.

First, there are two ways to hook up two water heaters: series and parallel.

SERIES

• In series, the water comes in to the first water heater, then passes through the second hot water heater, and finally goes to your fixtures.
• If you turn off the second water heater, then the water in the second one cools down between uses. This results in very long warmup times.
• If you want to turn one of these off, turn off the first one. (This is actually a legit way to save on energy costs if you don't need the extra hot water volume because in many places the water supply is colder than room temperature and the first tank allows the water to sit and reach room temperature before it's heated.)

PARALLEL

• In parallel, the incoming cold water is split between the two water heaters. Hot water coming out of the two heaters is combined and sent on to the fixtures.
• If you want to turn off one of the water heaters, you also have to close the valve to the inlet or the outlet. Otherwise you will just be mixing the cold water in one water heater with the hot water from the other heater.
• Leaving one heater closed off for a long time is a bad idea because the water can become stagnant and foul.

SECOND PROBLEM: Pilot light going out

Just turning off one heater should not cause the pilot light on the other to go out.

Low efficiency water heaters are low efficiency because their exhaust gases are very hot. Since the gases are hot, they will rise up a chimney flue with ease, but this also means you lose all that energy and warm air up your flue. Older heaters tend to be in older homes where there are enough drafts and air leakage to not worry too much about ventilation. One other thing: the hot air also carries also the moisture produced by combustion up and out the chimney.

High efficiency heaters try to reduce the temperature of the exhaust gases as much as possible in order to recover as much energy as possible. But as the exhaust gas is almost room temperature, it won't rise and thus has to be forcibly expelled. The moisture that is produced by combustion will also condense in the cooler air and will have to be drained out of a separate drain line.

High efficiency furnaces are also designed to be used in high efficiency houses that are well sealed, so you need to bring in fresh air lest you run the heater too lean and produce carbon monoxide. Another reason to bring in fresh outside (cold) air is that it is a waste to use the inside (warm) air for combustion and blow it outside the house.

Heat exchangers are used on the intake and outtake gases which further increase efficiency. This necessitates both intake and exhaust vents.

_{Thanks to commentator keshlam for additional material that was brought into the answer}