How wonderful, Olly! I do not think TV hurts your eyes, but I looked it up. Apparently watching tv in a dark room causes more eyestrain than watching tv in a room where theres some additional light. http://www.drweil.com/health-wellness/body-mind-spirit/vision/can-watching-tv-in-the-dark-hurt-your-eyes/

This article also says it’s not true that sitting too close to the tv can hurt children’s eyes. If children choose to sit very close to the tv it can mean they have visual-problems and need glasses. Meaning sitting close to the TV is an effect of poor eye sight, not a cause.

This of course applies to humans, we will just have to assume it’s similar with rabbits. Rabbits’ eyes are not quite as ours, and their field of vision is much wider.

As for radiation, I found info from FDA, the food and drug administration: Old TVs could emit low levels of radiation (x-rays). But LCD and plasma screens are incapable of that:

“It is important to note also that flat panel TVs incorporating Liquid Crystal Displays (LCD) or Plasma displays are not capable of emitting x-radiation. As such these products and are not subject to the FDA standard and do not pose a public health hazard.”

(The same article also says about old TVs that “It should be emphasized that most TV sets have not been found to give off any measurable level of radiation, and there is no evidence that radiation from TV sets has resulted in human injury.”)
http://www.fda.gov/radiation-emittingproducts/resourcesforyouradiationemittingproducts/ucm252764.htm

So Olly, you won’t be subject to harmful radiation from the TV. Neither will mum.