Will paper money become obsolete? I don't think so, and there are a few reasons why. It seems that while currency changes, there are always going to be uses for any currency. Gold used to be the standard, and after that paper money was the standard. Electronic banking will be the standard today, but there will be plenty of uses for paper money. Paper money will never completely go away, and like gold before it, there will be some niche use for paper money.
We don't have the gold standard, but gold is still very much in demand. Gold offers us a tangible good to put our money in when paper money loses value. When the economy is bad, or the dollar is weak, gold typically will rise. Gold is a good commodity for those who don't necessarily believe in paper money, and want to keep at least some of their assets in something that they know will always be around. Paper can be burned, or otherwise rendered worthless, while gold is always going to hold the same value.
Ironically enough though it seems that cash might be a safer way to pay for something than ever before. In this era of credit checks, and electronic banking, there is so much that can go wrong. If you have bad credit, or no credit, the only thing you may be able to pay for something with is cash. If you lose your credit card, you are out of luck, but you still have your cash that you can pay with. Paper money may be inflationary, but it is always a good backup.
If the electronic transfer lines crash, you can only pay with cash or credit. If you have no credit card, you can only pay with cash. Paper money is the only thing that you can pay with, and so the paper has a little bit of use in this case. Some people don't see the need for using a plastic card, and would prefer to carry around cash. While it can't be replaced if you lose it, some people just prefer paper money. It will always be around.
Paper money will always be a backup for when the electronic banking systems go down. Also, what if you don't want to charge something, or put it on your debit card. I was reading an article the other day, and the author made a great point that when you give someone paper money, you can feel it leaving you. There is no such emotion when you use a debit or a credit card. Using paper money can actually help us reign in our spending.
Electronic banking makes our lives easier, but cash will always be around. There are still some good uses for it, and there is no reason to think it will be obsolete anytime soon. Some like it, or can only pay with cash due to lack of credit, and some just prefer it the old fashioned way. Either way, there will always be a use for paper money.