Why a running toilet matters more than it seems
A toilet that refills every few minutes is not a cosmetic annoyance. A worn flapper passing water continuously can lose 150 to 200 gallons a day, which shows up on a North Carolina water bill within one cycle.
The sound also masks other problems. Once you stop noticing the toilet, you stop noticing the moment it starts overflowing.
The five minute dye test
Take the tank lid off and drop food coloring into the tank water. Do not flush. Wait fifteen minutes and look in the bowl.
Color in the bowl means water is leaking past the flapper at the bottom of the tank. No color in the bowl, but the tank still refilling, means the fill valve is not shutting off and water is going down the overflow tube instead.
What each result points to
A failed flapper is the most common cause by a wide margin. The rubber hardens, warps or gets coated in mineral deposits, and it stops seating. Replacement parts are inexpensive.
A fill valve that will not shut off is the second most common. Sometimes the float has been knocked out of adjustment; sometimes the valve diaphragm is worn.
A water level set above the overflow tube causes constant trickling. The waterline should sit about an inch below the top of the tube.
A cracked overflow tube or a flush valve seat that has corroded needs the whole flush valve replaced, which means pulling the tank.
When to stop and call
If you have replaced the flapper and it still runs, the flush valve seat is probably pitted and no flapper will seal against it.
If the toilet also rocks, leaks at the base, or overflows when other fixtures drain, the problem is not in the tank at all - it is the flange or the drain line, and that needs a plumber.