Your 2nd TV could be off frequency, especially for older analog CRT TV's and we are talking exclusively RF signals. In other words, if your TV loses the signal--no video and audio--the AFT (Automatic Fine Tuning--built in and always working in the background on TV's since the 1980's, no manual fine tuning ring around the round channel selector on 1970's and before models) keeps trying to tune to lock onto a frequency with a carrier. Essentially, it drifts into the neighboring channel as there is no signal on the channel it is supposed to be tuned to to keep locked on the correct frequency. Then when carrier returns to the channel your 2nd TV is supposed to tuned to, it is still locked onto the neighboring channel for video, but it is common for audio to lock properly while video is lost. This is really a rather common thing.
The solution is to re-tune to the channel original channel the 2nd TV is supposed to receive the Dish TV2 picture. This usually sets things right, but the AFT has now been pretty much set off frequency, which means that this will happen all over again once the TV loses video via RF. If you keep allowing this to occur, that is not set the TV to shut off before it loses video, the TV can become "permanently" off frequency (it will be stuck on the neighboring frequency and it won't tune back to the original RF no matter what you do), and you will have to have TV technician put your TV back on frequency.
This happens almost every night with my mom's old CRT in the next room being fed RF from Home Distribution, but, fortunately, the loss of video doesn't last that long and the AFT will lock on to the correct frequency or when it doesn't, we can re-tune to the original frequency with ease.
This is a problem ONLY for RF, not for the inputs of a TV. The inputs can have no signal on them and not be "ruined" by a lack of video and audio If you drift off frequency in the short 4 minutes that there is no video, then your AFT is already off frequency to a fair degree. In order for this to be a problem, the TV has to be tuned to a channel with no carrier for some hours OR--the big OR, dealing with loss of video--even in short periods of lost video--repeatedly over a LONG TIME. Loss of video for just a few minutes every day for 6 months to a year or more (depending on the TV's particular AFT) will add up to a problem as if you left on a channel with no carrier for many hours. Never, leave ANY TV on an analog RF channel, including an OTA "digital" frequency, which is still RF analog technology (as opposed to a Dish sat channel with no signal or black, for our purposes, that would be considered a digital channel) with no carrier whenever possible and one can avoid a tuner becoming off frequency.
Let us know what the problem ends up being.