If all of your lenses do this on the Rebel XTi/400D and none of them do it on another Canon EOS camera, then at first glance the issue would seem to be with the camera body.
The more I think about it, the more it seems to me the problem here has to be in the camera body, since voltage is being supplied to the lens even when the camera is turned off. It seems that perhaps the IS module and aperture mechanism are both also being constantly actuated. The noise in the video sounds exactly the same as an IS unit in one of my telephoto lenses did when it was going out. It would make the noise and rapidly vibrate the image in the viewfinder the way you describe.
Update: After some very careful tinkering around with a multimeter and an EOS body, I've discovered that there are some voltage differences between the pins on the camera-lens interface even when the body is powered down but a battery is inserted! They're not what one might expect, but they are there. For instance, the two A_GND (AF power ground) pins that are shorted together by the largest contact on the lens side when a lens is attached have a voltage difference of about 3 volts when no lens is attached. The Vbat (6V AF power) pin and the second A_GND pin have a similar difference (with the ground being the positive contact!) but the Vbat pin and the first A_NG pin have no voltage difference. There is also a voltage difference between the second A_NG pin and the D_GND (digital logic ground) pin.
This allows for the possibility that the problem is in the lens, rather than the body, even though the malfunction occurs when the camera is turned "off". It seems that when a battery is inserted, an EOS camera is never completely turned OFF. You can demonstrate this at home on any EOS camera: with a memory card in the camera and the memory card door closed, insert a charged battery into the camera and close the battery cover door. Notice that the memory card activity light comes on as the camera checks to see the status of the memory card.
I've never really considered it, but the IS "On-Off" switch on Canon EF lenses with IS may only send a signal to the body to send a command to the lens to park the lens' IS unit at the "home" position, rather than disconnect IS from the camera's control (in much the same way "focus-by-wire" designs work - there is no direct connection in the lens between the manual focusing ring and the the lens' focusing elements). This would simplify returning the IS unit to the optimal "home" position and parking it there when IS is turned off. It would also explain why certain EF lenses need to have IS turned "Off" before being disconnected from a powered up camera body, as Roger Cicala has notably reported.
Frankly, in 2020, repairing a Rebel XTi/400D that was discontinued in 2008 is almost certainly not going to happen at any price. Just opening the camera up will cost more in labor than its worth, and even sources of used parts salvaged from other unrepairable copies have probably long since dried up. If the issue is with the camera body, one would suspect a main board replacement would be needed since the power switching circuitry is embedded in the main board for that model. It's time to consider another camera.
If any other lenses you have do not do this on your Rebel XTi/400D and this lens does it on any Canon EOS body:
The lens' IS module and/or its main printed circuit board (PCB) is totally fried. I've had a lens that would occasionally do this intermittently, though only when the camera was powered up. It was becoming more frequent when I figured out, by looking through the viewfinder and seeing the entire frame "shaking" or "vibrating", what was going on. Replacing the IS module fixed the problem.
Send it to Canon for service or replace it, whichever is more economically feasible for you.
I would imagine replacing the IS unit in that lens would cost at least $200 including parts if the lens is out of warranty. Canon's minimum repair charges for labor tend to be dependent upon the lens' retail price (when it was a current lens). I've never sent a sub-$1,000 lens to Canon for repair, so I'm only guessing. The "standard labor charge" for my EF 70-200mm f/2.8 L IS II was $269 before any CPS discounts in both 2014 and 2017.
For your lens, which retailed for around $450-500 before it was replaced in 2016 by the EF 70-300mm f/4-5.6 IS II (currently selling from authorized dealers in the U.S. for $499), you can find new old stock for a little less than $400 and pretty good used copies for around $200.
Alternatives you might consider:
- Having your current lens repaired at a cost that is an appreciable percentage up to and possibly exceeding its current value (unless it is still in the warranty coverage period)
- Replacing it with another EF 70-300mm f/4-5.6 for around $200 used or $400 new (remaining old stock)
- Replacing it with the newer EF 70-300mm f/4-5.6 IS II that is pretty much the same in terms of optical performance but has faster and quieter AF for $499 new or about $200 for a very good used copy from a reputable dealer
- Replacing it with the APS-C only EF-S 55-250mm f/4-5.6 IS STM or EF 55-250mm f/4-5.6 IS II (older, pre-STM version) for between $80-160 used or $200-300 new
The EF-S 55-250mm f/4-5.6 IS STM, introduced in 2013, is a noticeable improvement over the older EF-S 55-250mm f/4-5.6 IS II, but it's fairly incremental. On an older crop sensor body like your old Rebel XTi/400D either one will give you about the same image quality as you older lens delivered.
The STM version will run about $300 new, while the older IS II is still available new from major retailers for a little less than $200. Used copies of the STM currently listed at B&H run between $130-160 for grades 8+ to 9+, while used copies of the older IS II are listed for $80-85 (8+ to 9).