Even though this is an old thread, I'll respond. I don't think wifi will cause much power drain either on or off. However, LTE might be a bigger power drain for the car because if you don't have signal the car/cell phone will constantly look to check if the signal is back and constantly try to ping a tower. Now you might think that a Tesla with a giant battery shouldn't really be an issue. Yes if it's just the cell draining the main battery but it's not. I think the cell drains the 12V and once that drains, the car has to wake up and recharge the 12V. Rinse repeat. Now I don't have concrete evidence of this but it could be possibility based on what I know about how Teslas work.
No, that is not how Teslas work. The car sleeps after about 10 minutes and doesn't ping anything. As soon as the car is asleep it will also shutdown the Wifi. It can only come alive if you wake it (then stays alive for 10 minutes) or if there is some BMS that needs calculation (I think the BMS thing is minimal) or the 12V needs charging. If you wake the car via the App it will send a push signal and it will either come (good LTE) or not come through (bad LTE) - the car will still continue to sleep if the LTE is bad.
The only thing that will keep the car alive on the connection end is an incoming software update. Usually Tesla pushes software updates over Wifi and very, very, very rarely over LTE, only when it is critical, at least here in Europe they don't anymore (I believe I only had one pushed to me over LTE over 1 year ago, they stopped doing that a while back)
So here is what happens if you have bad LTE - nothing.
There might bit a bit more strees on the LTE chip until the car is asleep, but that is minimal. I think you are comparing it to your phone, which has about 3000mAh total which in comparison is about 1000Wh(1kWh - 400V). The CPU of the car uses 250-300W or 300Wh in one hour.
So it will take one full battery charge of your phone in 10 minutes to make even a slight uptick on the car and you will still not notice it on the display (1000W for 10 minutes is about 100Wh. 1% on your display is 750Wh so about 1/7 th of % , not even showable...)
And we all know that even at worse Cell reception, the phones don't drain in 10 minutes from 100% to 0%. Which as explained above, will not even make a difference on a Tesla.
If you have bad Wifi or Wifi on - the car will check for updates and might start downloading one. If you have bad Wifi, it will take more time, more time means car is alive longer. And this will result in more drain. Not the bad signal on the Wifi itself, just the slow download.
For people who keep the checkbox in advanced and Tesla pushing updates almost daily, this results in more drain.
The drain really comes from only one place - the MCU which usses about 250-300W at idle.