Yes, too much visine is bad for your eyes. So you should not use it too much. As we know that visine is an ocular decongestant, which is used to get the red out by constricting the blood vessels in the conjunctiva. Then the problem is that with chronic use, you can get rebound redness. For dry eye it's better to use a lubricating drop like Refresh or Thera-tears that basically just replaces your tears. Anyway, using it too often, or too long can irate your eyes and damage your eyes too. So just be careful about it.
I think the using of visine eye drops is not bad for eyes in some special occasion. For example, if you are going to join a job interview but your eyes appear red. Proper using some visine eye drops will help you appear good. But the excessive use of visine eye drops or similar medicine is not good for eyes. They just temporarily mask the redness in your eyes by constricting the blood flow. So, be prudent to use visine eye drops.
No, watching TV too close will not damage your eyes. While many people like your mother say it does harm to your eyes when sitting closer than necessary to the television, but the truth is that it will not damage your eyes. Generally speaking, sitting too close to watch TV may give you a headache, but it will not damage your vision. Because Modern TVs do not emit harmful radiation, so eye damage due to radiation is also not an issue. As with sitting too close to the television, you may get a headache from reading in the dark and eye strain, but it will not weaken your sight. Next time, when your mother tells you again, I think you can tell her.