So I think a lot of us have been told by our parents that sitting too close to the TV is bad for your eyes. Is there any truth to this? when and where did this idea begin?
I have no real answer. I sit very close to a laptop and second monitor all day, and my eyesight is fine.
I think this belief stems from cathode ray tube TVs and the possibility that they were emitting excess electrons, alongside a complete misunderstanding of how CRT works.
Certainly my parents were concerned about it, but no one seems too concerned about LCD screens in office (or, back in the 80s/90s) CRT screens just 50 cm from your face.
I’ve also wondered about this. There are definitely people my age that still believe this (specifically you should be no closer to the TV than the diagonal size of the screen).
My feeling is is that, if it was ever a valid concern, only applied to older CRT TVs. But I’ve no evidence for that.
There were a few memes going round in the VR community to the effect that your parents made you sit away from the TV, but nowadays you strap a VR headset (which is basically a TV) directly in front of your eyes.
It began in the 1960s with a manufacturing defect in several GE model TVs. The problem was a misaligned shield over a vacuum tube which resulted in dangerous levels of X-rays being emitted, mostly at a downward angle. If you were sitting on your couch, no biggie. But if you were a kid sitting on the floor close to the TV, you were right in the path of the dangerous levels of X-rays.
GE fixed the manufacturing problem, and issued a widely publicized recall to try to get all of the dangerous TVs fixed. Several thousand of the dangerous TVs were never tracked down though.
Even though the problem was fixed (except of course for the TVs that were never tracked down and repaired), people now had the idea in their head that sitting close to a TV was bad. This urban legend was fueled somewhat by the fact that the old CRT type TVs did in fact emit low levels of X-rays from the CRT, probably not enough to be worried about even if you sat very close to the TV, but there you go.
Modern TVs and VR displays do not emit X-rays.
There was some talk about CRTs back in the day, but it was mostly ignored. LCD screens don’t emit X-rays so there never has been an issue there.
ETA:
There have been some studies that indicate constantly focusing on close things when you are young (computer monitor, always sitting close to the TV, reading a lot) can lead to myopia. I read a lot as a kid and started spending a lot of time looking at a computer monitor starting when I was in high school, and I am myopic. So there is my sample size of 1 to confirm the studies.
The Straight Dope:
https://www.straightdope.com/21342032/will-sitting-too-close-to-the-tv-ruin-your-eyes
The Sci-show looked at that Inuit study and more recent ones, it looks like it is more related to having kids having less exposure to the outdoors and the sun rather than just studying or looking at screens or books closely.
To tease out the effect of cultural environment, Australian researchers from the University of Sydney looked at 6 and 7 year old ethnic Chinese children living in Sydney and Singapore. The kids’ parents had similar rates of nearsightedness–around 70%–in both study groups. But in the kids themselves, the difference was stark. Only 3.3% of kids in the Australian group were nearsighted, compared to 29.1% in Singapore. And the children in Sydney actually did more near-work activities, like reading and homework, than the kids in Singapore, so that couldn’t possibly be the cause.
The only difference between the two groups of children that could account for the difference in myopia was how much time they spent outside. The kids in Sydney spent more than 13 hours a week outside, the kids in Singapore only 3.
The warning goes way back to early television.
A syndicated article from late 1949 gives the advice that “Sitting too close to the television set presents a visual problem. To avoid seeing the pulsing and unpleasant flicker and flutter caused by electrical disturbances, sit at least eight feet from the screen.”
Do’s and don’t for watching television in 1950. "Sitting too close to the screen and watching television in the dark are unfortunately habits which many parents, as well as children, have acquired.
The Atlanta Constitution is 1951 wrote, “And remember too that children often need to be reminded not to sit too close to a television screen,… Sitting too close or viewing television in the dark is apt to cause eye fatigue.”
A 1954 Small World cartoon has a young brother tying his littler brother to a chair to keep him from sitting too close to the television.
The New York Daily News in 1956 ran an article from psychologists saying that tv is good learning but reminded parents “to keep children from sitting too close to the television set.”
It’s hard to tell how many different articles are represented because of syndication, but my guess is lots and lots before 1960. The flickering aspect arises around 1949 with eye fatigue coming to the fore in 1950. I sorted by relevance because of the volume of hits, so onset dates are approximate.
Damn! I could have stated, without any doubt whatsoever, that this warning could be directly attributed to my mother! 
There’s a whole category of (now collectible) “TV lamps” so the viewing room doesn’t have to be totally dark. Some were quite exquisite, while others were, well, just small lamps.
More children have become nearsighted over time. So whatever is new tends to be blamed. As stated above, lack of sun exposure is now thought to be a major myopia factor.