I think most humorists (writers, speakers, actors, etc.) and experienced teachers would probably admit that humor is a difficult quality in writing and in life to teach and to learn. At its core, the best humor is usually somewhat innate. "Wit" as referred to in your question of writing a witty article, is a specific category of humor that might described as intelligent or clever. Most often, someone who is described as "witty" has an ability to make socially relevant jokes quickly, and in the best context to get a laugh.
If you are not a naturally witty person, you might never become a witty writer, but certainly there are things you can do practice and develop this ability. First, like any other skill, you should take advice from the pros. This means read, read, read. Read other witty articles. Often wit presents itself in the form of a voice, and hearing or reading that voice over and and over can help you use something similar in your own writing.
Because wit is a form of intelligence, it is easiest to be witty about subjects you know very well. It has been said that the smartest citizens in a country are its comedians. You must really understand a subject inside and out in order to be able to make fun of it. Additionally, the best humorists also know their audience very well. Writing a witty article requires an acute understanding of both your subject and your audience.
Another tip for writing witty articles is to choose subjects that are well known to a wide audience, but instead of writing (or saying) what everyone else is, put a new twist on them. One of the most popular sources for witty articles today is The Onion, a satirical newspaper published both online and in print. The Onion's success rides on keeping its articles current and pop cultural. Writers often choose very well known stories as subjects, then tell them with an unexpected twist and a high level of hyperbole. The humor then lies in the fact that though the story is pretty well known, it is being told in a different (and often inappropriate) way.
A final tip for writing witty articles is to have a very strong opinion about something. Mark Twain is one of the most well known American humorists. He wrote satire and irony, predominantly, and poked fun at some of the most hot button issues of his time, namely slavery. It was easy for Twain to poke fun but also speak very clear political messages in his writing because slavery was something he knew very well, and clearly disagreed with.
Humor writing is very vulnerable writing. You must be prepared to be ignored, rejected, or even considered offensive. Above everything, it is a skill that gets easier and better with practice. Good luck.