Is there any evidence that watching television for too long or while sitting too close weakens the eyes (leading to myopia, for example) ?
There are two theories about this myth which I've heard:
Let's debunk them in order
Eye strain theory
It's a well known fact that watching TV causes strain (tiredness) in the eyes. Any ergonomics source will tell you that. The problem here is that eye strain causes are many - TV, computer, books, driving, etc.
So, yes, TV can cause eye strain, as many different other things - this partially debunks the myth because TVs don't have any particular status here.
Does eye strain cause myopia? No. Eye strain might precipitate the onset in some cases, but there are no ways to prevent myopia. Furthermore, it's quite certain that myopia is of genetic origin. See the following references: 1, 2, 3, 4
So we can see that the eye strain theory does not stand on its feet.
Cathode rays theory
This is easily debunked. Yes, old school CR tubes did use beams of electrons to generate images. The thing is, these beams are tightly focused on the screen in order to have a focused image. This is obtained through different means, but basically the electrons hit phosphors in the screen which is a wire mesh - conveniently it is also a Faraday cage which prevents them from going anywhere.
The actual image is made of phosphor generated light. It's just light :-)
That said - there is a small amount of X-rays being emitted, but it's tightly regulated to super-low levels and it is widely considered innocuous.
According to the Straight Dope, in ordinary circumstances watching too much TV or sitting too close will only lead to eye fatigue, not eye damage. Some studies claimed that exposure at too young an age can cause myopia, but at the time of writing (1992), there was no definite proof.
Focusing on anything for an extended period of time can cause eye fatigue. Older eyes tend to recover from the fatigue more slowly, which may lead to some perceived vision loss during the recovery time.
There's no real evidence to suggest focusing on a TV is any worse than anything else.