I've heard a story that a weak vehicle battery can destroy an alternator or cause an alternator to fail early.
Is this true, or is it largely fiction?
Your question is difficult to answer without you knowing the inner workings of an alternator.
The basics. If you move a magnetic field near a coil of wire the electrons in the wire get excited and electricity will be made. The amount of electricity made depends on the size of the magnetic field and its speed. The bigger the field and the faster it's moving the more electricity is made. Also if you have a coil of wire and you pass a current through it it will create a magnetic field.
An alternator has 4 basic components; rotor, stator, voltage regulator and bridge rectifier.
The whole system is designed to do two things. First, refill the battery after it cranks the engine. Second, supply power to the rest of the car. The way the whole thing works together is that the voltage regulator senses the voltage of the system and adjust the rotor current accordingly. When for example the head lights are flipped on this presents a greater load and lowers the system voltage. The voltage regulator senses this and adjust the rotor current accordingly. Then you go to pass someone on the highway and mash down the gas pedal. This speeds up the engine and the system voltage will go up. The voltage regulator lowers the rotor current to bring the system voltage down. This cat-and-mouse game constantly goes on in the charging system.
When an alternator is rated for a particular current output, lets say 100 amps, that rating is at 2000 RPM. The alternator can comfortably make 100A at 2000 RPM. It is designed that way because typical cruise is around 2000 RPM. At idle, the rotor is spinning slower and the alternator is incapable of making it's full rated current. At idle, is when a charging system can get into trouble.
A battery is a pig. A battery will take all the current it wants and no less. The current it want is proportional to its state of charge. A discharged or weak battery is very hungry for current.
To bring it all together, when a car has a weak battery that battery will want lots of current. The current demand of the battery lowers the system voltage so the voltage regulator compensates by sending more current through the rotor. At idle, the alternator is incapable of making the needed current. Because of this the system voltage drops even more and the voltage regulator sends the maximum current through the rotor.
This maximum-load, minimum-speed condition is where the wear happens. At the minimum speed the minimum amount of cooling is available from the built-in fan. At the maximum load the voltage regulator will push the maximum amount of current through the rotor and through the brushes and slip rings. The brushes and slip rings get hot and with no additional cooling available from the fan they will wear faster.
If the RPM is increased to above 2000 the situation gets better because more cooling is available and the current through the rotor decreases. This will unfortunately move the wear point from the brushes to the bridge rectifier because now it has to rectify the maximum current. This is preferable, however, as the bridge rectifier is a solid state component and is much less subject to wear.
Absolutely yes. I sell alternators all day every day - and a poor battery is often the root cause of alternator failure.
A battery with a short in it will cause the alternator to run at full output for extended periods of time, if not continuously - and they are not built to do this. Alternators the world over are built to supply an initial high current, tapering back as the current used to start the vehicle is replenished. Constant running at full output simply overheats them and the rectifiers fail. *See edit below
If your battery is 'Open Circuit' the alternator will either not start charging at all or will erratically bounce from low to high voltage. This causes the regulator to fail prematurely (or immediately!)
A battery that simply will not accept charge won't necessarily cause premature failure of the alternator unless it is permanently at a low voltage, in which case see the 'shorted' section above.
Hope that helps?
*EDIT: Some bigger commercial and marine alternators are designed to run at full output constantly, but I don't get the feeling you're talking about those :) In 16 years of selling parts for alternators I've never heard of the brushes and slipring wearing and the grease in the bearings overheating from high charge current at idle. These issues are caused by other problems not in scope of this question.
A battery that has an internal electrical short, usually one of the plates has come loose and is touching its neighboring plate will cause the alternator to work much harder than normal. This can shorten the life of the alternator. This condition is usually discovered quickly as the battery will not function well in this case.
When alternators or motors work harder they produce more heat. The alternator damage in this scenario is caused by heat. The insulation on the rotor windings can be damaged. The grease in the bearings can be overheated to name a couple of problems. Most alternators are designed to produce the maximum rated power for only short periods. The system design is for the alternator to charge the battery for a less than five minutes and then drop to a charge rate of less than 10% of its maximum rated power.
In order for an an alternator to function it requires a functional battery. The battery current is required to required to energize the field coil which produces the magnetic field required for the alternator to produce energy. The alternator sends 5 amp-hours at about 3000 RPM to the battery and the balance of its output goes all the other systems in the automobile. The primary function of a car battery is to start the vehicle. Once started then the alternator takes over completely. Inside the alternator there are a series of diodes which convert the AC current into rectified DC current. Sometimes these diodes fail and pass AC current into the regulator and subsequently destroying the regulator. The Battery acts like a very large capacity capacitor. If DC is applied then it stores energy. When AC is applied to a capacitor in parallel it acts like a short circuit destroying the regulator further and cooking the alternator coil winding. The also has the effect of potential seriously destroying the battery as well if left unrepaired.