Flapper leak
A warped or scaled flapper lets water seep into the bowl, so the tank keeps topping up.
The tank hisses, refills every few minutes or never stops filling after a flush.
Short answer
A toilet that keeps running is almost always a tank problem: the flapper is not sealing, the chain is catching, or the fill valve is set too high and water spills into the overflow tube. All three can be checked by lifting the tank lid. None of them is a reason to replace the toilet.

A warped or scaled flapper lets water seep into the bowl, so the tank keeps topping up.
The flapper cannot drop fully closed after the flush.
The float lets the tank fill above the overflow tube, so water runs into the bowl.
Scale or wear keeps the valve from shutting off, causing a steady hiss.
Running toilets are easy to ignore because the sound fades into the background. In South Florida homes, mineral scale is a common reason fill valves and flappers wear out before their time. A plumber can replace both in one visit, check the supply line and valve, and look at whether the rest of the toilet is sound. If the tank is old enough that the flush valve seat is pitted, a new flapper will not seal for long. A plumber can replace the flush valve or the whole tank kit in one visit.
Call when a new flapper does not stop it, the fill valve hisses or will not adjust, the shut-off valve will not close, or water is leaking anywhere outside the tank.
A steady leak can waste a large amount over a month. Your water bill or meter will often show it.
That is a slow flapper leak. The tank drops, the fill valve tops it up, then stops.
Many homeowners do. Close the valve, flush, unhook the old flapper and match the size, usually 2 or 3 inches.
The chain or handle is catching, so the flapper does not drop. Adjust the chain length.
Usually the fill valve. It is not closing fully, often because of scale or wear.
They can wear flappers and seals faster. Many toilet makers advise against them.
If it runs constantly and you cannot fix it yet, closing the valve between uses saves water.
Rarely. The tank parts are inexpensive and replaceable on almost every toilet.
Yes. Sewer charges are usually based on water use, so a running toilet raises both.
The flush valve seat may be scaled or chipped. A seat repair kit or new flush valve can fix it.