I recently installed a new gas water heater with factory installed heat traps. Everything works fine, but the other day I noticed that the cold water supply was warm to the touch. This only happens when the burner is on, and the pipe is warm for only the first three or four inches coming from the heater. It is cool the rest of the time. Is this normal when the burner is on? Could I have a defective heat trap as it should be blocking this flow? The hot water heat trap appears to be working properly.

You never want anything completely blocking the inlet. As the cold water, that replaces the hot you draw, heats, it expands. The excess will go somewhere, if not out the inlet, the pressure will go up until it creates an opening, if all goes well the relief valve.

This as created problems where a public water system has added back flow preventers without the homeowner adding an expansion tank.

So a little hot water going out the inlet is a solution, not a problem. I don't know if it is worth it to you to insulate it or not. It is to me, but where my water heater is, I need to insulate the line to keep it from freezing.