What wires were the fan (or in your case the fan remote) originally connected to? The black on the left and the two neutrals in photo one?

IF the capped off black (wire on the right) leads to your switch box, then I'm going to GUESS that you do have a switch loop, but one that was incorrectly wired and not being used by the current fan because it has a remote, which only requires a constant hot to function. In your case, it appears that the switch loop was switching the neutral and not the hot wire.

I'm making assumptions based on what I see in the photos, wires need to be traced out to prove or disprove what I've stated.

I'm puzzled by the fact that your voltage tester beeped on both black wires. That suggests to me that you might not actually have a switch loop there, but rather two hot cables in the same box.

Is it possible that one cable comes from the switch you're using now, and the other is from some other switch somewhere? Or even live all the time?

Normally a fan and light combination will be wired with 3-conductor cable (red-black-white) with the light switch feeding power to black and the fan switch feeding red (or vice versa). White is the common neutral. But AFAIK it wouldn't be improper to have two separate cables coming from two separate switches, or one switched and one always hot.

Non-contact testers are useful, but I don't fully trust them. I'd test those wires with something that actually draws some current. I like using a 15 watt or 25 watt incandescent bulb in a rubber pigtail socket for this. Adding an insulated alligator clip to the white pigtail wire is helpful.

A neon glow tester would be OK as a substitute.

Just be careful when testing with the power on, of course.

Voltage testers are known for false positive. If it's connected to an open switch and parallel to a bunch of other wires, it may not have been actually hot.

Get a better tester or use a voltmeter/multimeter on volts ac with prongs on black and whites to see which is live. I recommend the klein tools tester. I have also had a cheap brand one in the past that before I threw it away, would beep even when I had it held away from everything so yes they do go bad.

Thanks for the replies, everyone. It is certainly possible I'm getting false positives with this voltage tester. For what it's worth, it is brand new, as I purchased it exactly for this reason. The one I bought is linked below:

https://www.amazon.com/Non-Contact-Electrical-Inductive-Multimeter-PM8908C/dp/B01AFQOM2U/ref=sr_1_2?ie=UTF8&qid=1508452056&sr=8-2&keywords=PM8908C

To double check my initial readings, I turned off the power and flipped on the light, and none of the four resulted in any beeps. I then turned back on the power, left the light switch on, and I got strong beeping/light flashing when just touching both outside (black) wires pictured in the first picture in my initial post. The white wires that are braided together don't give off any beep, and the bare wires that are braided together and secured with electrical tape result in some beeping and lights flashing but not as pronounced as the black wires.

I did manage to find the directions for this Harbor Breeze fan, and if I could safely figure out which wires go where, I'm reasonably confident I could make the below connections and get the fan back up:

However, it sounds like what I bought isn't enough to accurately (safely) make that determination. If anyone else has any additional ideas how to identify my wires in accordance with the above directions, I'd certainly be interested in hearing them. Otherwise, I'll reluctantly have to hire an electrician to put this back up, as I really don't do enough work with electrical wiring to justify spending more money on a voltage tester. Thanks again.

I have a Harbor Breeze fan, with similar instructions. The receiver's red wire is connected to the power source, and its black and blue wires go to the fan's motor and light (I'm not sure which is which, from your description). The white and ground wires just go to the equivalent wires from the supply. The problem I had is (was) that my electrician had run separate wires from the wall switches to the fan box, expecting that the fan would be controlled by these switches. I tried to bypass the receiver in the fan, but could not get a wire down to the light (I didn't want to use the remote control). So, now, I have one wall switch and control the lamp brightness and the fan speed and direction all with the remote. I make these settings, then turn off and on the fan with the wall switch. I only need the remote when I change the settings.

If you have a supply wire (black, from your picture) which is on or off depending on a wall switch, then you can use it like I did. If you want to use only the remote, then find a black wire which is always on.

You need to figure out where all the black wires go. One is probably an always-on supply wire. One may be a switched supply wire, and one (which now tests no voltage), probably goes to another light or a receptacle. For testing this type of wiring, I use an incandescent bulb in a socket which has individual wires, like this.

Thank you, mtvhike - that was very helpful. So if I were to purchase one of those inexpensive bulbs that you provided a picture for, I would hook up the black and white wires to the corresponding wires from the box? Or is the white wire not even needed for the purpose of identifying the wires? Finally, how would I determine which white and ground wire to use from the box since there are two of each, or is it safe to use both of them with the corresponding wire from the fan (as was previously done)?

The bulb in this socket draws enough current that the spurious voltages will not occur. I assume that there are two black/white pairs which come from the breaker box. It's easy to see which black is which. First, connect the white on this socket to the twisted whites and touch the black on the test light to each of the blacks and see if any of the blacks are hot. Once you have confirmed that at least one white/black pair is powered, turn off that breaker and make these temporary connections more secure (not just touching live wires). Turn the breaker back on and then turn on and off the likely wall switches. If none of the switches make the test light go off, then you have found a source of power. Repeat the above for all the black wires and see if any of them are switched. Finally, and I'm not sure about the following test, but if one of the black wires never shows power, it must go to another light fixture or receptacle. Connect your test light between the black power source and this unknown black wire. If your test light glows, then that black wire goes to something which draws current (perhaps a ceiling light with an incandescent bulb in it), and you can figure out what that is by observation. If it doesn't glow, then take another test light, put a regular plug on it, and plug it into each of the likely receptacles. If this second test light glows, then the two lights are in series and you know where this unknown wire goes and you can nut it to the power source wire (or to the switched wire, if you want that receptacle switched).

One thing I don't know is how to trace out the white wires. If each of them act as a reliable neutral, then maybe it doesn't matter. If it does matter, I don't know how to identify which is which.

Sorry for this long response; I hope I'm clear.