This is kind of like those washing machine instructions where they tell you before calling service to be sure the machine is plugged in, but: Are you sure that the noise is coming from the toilet? When you open the top and put your head down next to the fill valve, do you hear noise right there? If the fill valve is letting water in ("running nonstop"), then there will be water coming out of the refill hose as well and into the overflow riser. Is there water running out the refill hose and into the overflow riser? Is the end of the refill hose "daylighted" above the lip of the overflow riser?
If you turn off the water to the toilet, does the "running" noise stop? If so, if you leave the water turned off for a couple of hours, does the level in the tank fall? (Mark it with a pencil.)
A while back, we had a guy who was experiencing a fall in his tank water level when he turned off the water, and the toilet was running regularly to refill the lost water. He put dye in the tank but none ran into the bowl. He was mystified. Ghosts? Nope. Turns out that he put the dye in the tank but he didn't mix it up so all the water in the tank was that color. The problem was that he had a crack in the overflow riser about two inches from the top of the riser. It was hidden under the clip on which you clip the refill hose. Water would run through that crack and down the overflow riser, causing the toilet to "run"/refill. Because the dye was at the bottom of the tank, not up where the crack was, the water running into the tank through the overflow riser crack was clear. Much better to mark the water level with a pencil and see if it falls when you turn off the water; that way you will at least know that water is escaping the tank by some route -- or not.