In 1947 India/Pakistan left the British Empire. They were no longer going to be run under any British government rule, and would adopt their own constitutions. Though it wasn't as easy as simply moving one government out and moving a local government in. One of the biggest problems was that India at the time housed two major religions, Hindu and Muslim. It was decided that the best way to deal with this would be to separate these two religious groups into two separate countries, India and Pakistan. So it was decided that the Islamic people, mostly living in the area now known as Pakistan would get their own country. The rest of the India would go to the Hindu people.
Not everybody was happy with this situation, in particular there was a problem in the state of Kashmir. Kashmir was being run by a Hindu, Maharaja Hari Singh. He was undecided about how he wanted to go, and delayed the decision to join Pakistan or India (possibly with the hope of remaining independent from both). With more and more pressure building up he eventually gave in and in exchange for military help joined India. Needless to say many people were not happy with the decision, particularly since a great deal of the population of Kashmir were Muslim. India legally holds Kashmir, as one of their states, however Pakistan believes that the decision was not fair and that Kashmir is rightfully Pakistani. To this day whether or not this state should remain part of India is hotly contested.
This would firmly fall under one of those borders your probably shouldn't actually visit as there are still regular attacks and bombings. It's unfortunate as it's one of the more beautiful regions and used to be a hub for tourism.
The Green Line, Cyprus
Why the Greek Cypriots and the Turks don't get along changes depending on whom you listen to. The important part of the history of the line dividing this small island nation, sometimes called the Green Line, is that one side is Turkish-Cypriot, and the other side is Greek-Cypriot. The line itself is actually fairly wide in some areas, and residing in this “line” are the United Nations Peacekeepers. For the most part this space has remained untouched since the Turks landed on Cyprus in 1974 to support the Turkish-Cypriots. Within the buffer zone, remnants of this military intervention are airplanes left at the old airport, kitchen tables in homes left untouched, and an abandoned auto dealership with brand new 1974 cars still waiting to be sold.
Unlike many of the other non-borders that were separated by war, this one is not only accessible but also very safe to visit. Parts of it are a little difficult to see and might be more protected from the public. The best place to see the Green Line is in the capital of Cyprus, Nicosia, since it is divided right down the middle. The division line has turned into a bit of a tourist attraction, and is fairly accessible by just walking down a few small streets. There are still military installations along the border, so taking pictures in many places is not allowed. You can even walk across the Green Line and visit either side. The beauty of exploring this part of the city is the fact that you can still peer into the untouched areas and see parts of 1974 Nicosia.
Hans Island, Canada/Denmark
To many people from Denmark and Canada, Hans Island is a regular feature in the press. It's a 1.3 square kilometre (half a square mile) rock found in the Kennedy Channel of the Nares Strait that divides Greenland from Canada. It's best described as a large piece of granite in an icy river with a lot of bird poop on it. While the occasional news story comes up about this “disputed” land, Denmark and Canada are keeping things low-key, and universities from both countries have conducted joint scientific studies. Canada and Denmark (as Greenland is still technically part of the Kingdom of Denmark) are not at war over this island that lands right in the middle of their maritime boundary, because except for a few scientists, this island has no real value at all. Occasionally news articles are written on the subject, but it appears to be a border dispute that may not be worth resolving.
There are unsubstantiated stories of boats leaving bottles of peach schnapps and Canadian Club whiskey along with their countries’ flags and notes, attempting to claim the land. While these ideas are fun, they're not really legally binding. Visiting Hans Island is not really a good idea, mostly for the fact that it's very remote, surrounded by ice, and not very interesting.
Mont Blanc France/Italy
One of the real challenges of marking borders is geography. Land where borders fall is rarely flat (sometimes it's not even really land), so when it comes to drawing lines on bumpy areas it gets tricky. It gets really tricky when the land is very mountainous, and gets very political when the mountain in question is the tallest mountain in Europe. Mont Blanc (or as the Italians call it Monte Bianco) is situated between Italy and France. If you side with the French government (and many of the textbooks I've seen) it's within France. However the Italian government disagrees, and there have been several campaigns to say that Mont Blanc is in fact right down the middle.
Mountaineers from both countries have weighed in on the debate, many with different opinions. There is even a story of heavy lifting equipment attempting to move the peak towards a particular border. This border is getting easier to get to because both governments not only want to claim it, but also want to turn it into a travel destination. Cable cars and mountain excursion abound with the hopes of showing you the highest peak in France... or Italy... well Europe for sure.
Hong Kong
This is a pretty complicated situation. Up until 1997 Hong Kong was a British colony, and pretty much its own country although run by a Governor. After 1997 Hong Kong became part of China... sort of. Technically it's part of China, however they've kept their own currency, official languages, and China even still has an embassy in Hong Kong. Hong Kong also has some civil liberties protections in their own constitution. Things like freedom of speech and the ability to live as capitalists are rights given to the citizens (?) of Hong Kong. The two problems with this are that this constitution expires in 2047, and many argue that the people of Hong Kong had no say in any of these decisions.
As far as a place to visit, it's open to tourists, is fairly safe, and is considered a world-class city. While technically you are on Chinese soil, there are enough caveats to this that you can also say you're not really in China either.