Everytime your skin burns from exposure to the sun increases your chance of developing skin cancer. UV radiation from the sun can cause the DNA in your skin to mutate; if the mutation occurs in some gene regulating growth of the cell, it may result in cancer. Over time, these mutations can accumulate, leading to a greater chance that it will develop cancer. Since children have a longer time for these mutations to accumulate, it becomes especially important to protect your children from harmful UV rays. So yes sunscreen is important, esp. in the summer. They have new sprayable sunblock which makes it easier to apply the sunscreen to your kids. Or there are the kind with different colors that kids like to use (it's like they get to paint themselves with different colors).
Now it may be a little overboard if you completely cover them (with sunblock, hats, sunglasses, special clothing, etc.) everytime they go out. We actually do need a little exposure to the sun; the sun's exposure relates to the amount of Vitamin D your body takes in as well as the amount of melanin (the dark pigment in your skin responsible for a tan which naturally aids in protecting your skin from UV radiation) that your body produces. So I'd say, if your kids are going to just go out to ride their bikes for a few hours etc. then you should worry about making them "sunproof", but on the other hand, if you're going out to the beach for the day with your kids, then apply some sunblock to them.
As for sunscreen being "bad" for your liver, anything foreign that enters the body (including being absorbed through the skin) goes to the liver to be processes. The liver is your filtration system, removing toxins that enter the body. So it is the liver's job to process these "bad" toxins; however, it doesn't mean that it is harmful to the liver itself. Usually liver damage is a result of prolonged exposure to a particular toxin, esp. if there is not enough time between exposures for the liver to recuperate. Alcoholics that drink everyday may develop liver damage because of the continuous exposure to alcohol. You also have to consider the means of entry into the body... Inhaled toxins enter the blood the fastest followed by ingestion. The poorest method for an item to enter the body is by absorption since the skin creates a natural barrier to the outside world. So I think that the amount of any toxins present in sunblock that would reach the liver would be so minute as to cause little to no damage to the liver. I'd be more concerned with the sun's exposure resulting in skin cancer later in life rather than the possibility of some liver damage from sunblock.