There is an overhead ceiling fan with no wall switch, it is turned on and off by pulling the chains. When running a wall switch, I ran romex cable from the wall to the fan, connected the black hot at the fan to the black in the romex, and taped both ends of the white with black tape to indicate they are hot. When I flipped the breaker back on, before installing the fan, using an electrical tester, all three, ground included, in the romex sheath indicated they were hot. If you touched, very carefully, the white with black tape or the ground, nothing was felt. I installed the fan and it works normally. I used two different electrical testers, the ones that beep with current. First time I ever had all three wires in romex indicate hot, is this normal or is there another issue I should check?

A switch loop is wired this way. The ceiling BLACK goes to the WHITE of the switch wire and then the BLACK of the switch wire goes to the BLACK of the motor. Mark the current carrying WHITE with black tape. The ceiling WHITE goes to the motor white. Grounds are all tied together.

A switch loop is wired this way. The ceiling BLACK goes to the WHITE of the switch wire and then the BLACK of the switch wire goes to the BLACK of the motor. Mark the current carrying WHITE with black tape. The ceiling WHITE goes to the motor white. Grounds are all tied together.

Is it crucial that the black ceiling go to the white switch? Can it go to the black switch, and have the white with black tape go to the black on the motor?

It should be an easy fix with a switch box and a ceiling box. Crucial? No. Black return is code but many people use white with tape or marker. Should make no difference. If you have hot on all, you wired something wrong.

You are using the wrong type of testing device.

A switch interrupts the hot feed to a device, so the two insulated conductors would only be hot. With the switch in the OFF position, the power would only be present on the white feed to the switch. With the switch in the ON position, both conductors would be hot.

It sounds like you are using a proximity tester. Get a meter! Proximity testers are designed to pick up any presence of electricity. Because both conductors are in the same cable, the testers you are using pick up its presence.

A meter, however, allows you to test individual conductors. In this case, you would put the meter on ACv and place a probe on the black return conductor. The second probe would go to the white conductor. With the switch turned OFF, there should be a 120v reading on the meter. With the switch turned on, there should be a 0 volts. Meters measure the difference between the two conductors. When the switch is turned on, both conductors are the same, so thee is 0 (zero) difference.