I have a standalone bathtub that I never use. When I turn on the hot water tap, black water comes out in surges. It actually seems to get worse the longer I leave it on (it may lessen after a while, but it would be a long while).

Today I had it on for a few minutes, and then I turned on the hot water in the sink next to tub and it had the same problem (I have double sinks and this was the lesser used of the two; the other one didn't have the problem).

I haven't noticed a smell, so if there is one, it's not strong. I also haven't collected the water in a glass to see if the black stuff falls to the bottom (I'll do a sniff & sediment test later tonight). Also, it's city water, and this happens every time I turn on the hot water tap in the tub (which, admittedly, isn't often). The hot water heater (like my house) is from 2002; pipes are PVC.

It seems like this would be a hot water heater problem, but I don't notice the black water in frequently used taps.

Any help or ideas on what it could be and how it can be permanently remedied would be appreciated. Thanks!

You need to clarify something.
The house is six years old?
The supply pipes are PEX not PVC or copper?
City water or well?

You need to check a few things.
1. There is a drain valve on the bottom of the hot water heater. Get a bucket and drain some water into it. Check to see if more of this black stuff comes out of the bottom of the heater. What happens when you do this? If more of this comes out there, continue to do this until the stuff stops coming out. (leave the water on to the tank while doing this) Be careful as the water will be hot!
2. On the kitchen sink remove the strainer on the end of the tap and check if its filled with this black stuff. What did you find?

There are two reasons why this can occur. The first being that the pipes that feed the tub, Assuming its a free standing one like you said are metal where they enter into the faucet. They could be rusting. Same goes with the faucet used on the tub. But the faucet and pipe would not explain why it takes a long time to clear. As often a few seconds clear water comes out once the faucet is flushed.

My bet its the hot water heater. The reason is that the amount of water that is drawn out of the hot water heater is often less with a regular faucet running then when a tub faucet is used. This large volume coming out of the tub faucet can pull the sediment up out of the bottom of the hot water heater while a regular faucet would not.

To answer why the hot water heater could be the cause is twofold. Township and well water contains some sediment. As the water comes into the tank the suff settles to the bottom of the tank. This happens a lot in towns that often flush their fire plugs along the street. They do this to remove sediment from the street pipes. When they do this some of the dirt moves into your house system where it settles in the hot water heater. Also a failing hot water heater can be the cause as well. Builders oftentimes use cheep heaters that can fail fairly quickly. Be prepared to replace it soon. If you hear a grumbling sound coming from the heater when it operates that is because of the build up of sediment on the bottom. Another sign of a soon to fail hot water heater.