From time to time, New York City's Metropolitan Transit Authority has claimed that the Lexington Avenue line in Manhattan, where 4, 5 and 6 IRT division trains run on parallel tracks from their starting points in various parts of Manhattan, the Bronx and Brooklyn is "the busiest subway line in the world".
The only direct reference to this claim that I could find in a search was an article in the New York Times about the fatal train derailment in August, 1991 near the 14th. Street Union Square station that closed the line in Manhattan for 6 days. A link to the article is posted below. It says that, at the time of the accident, the Lexington Avenue Line was moving 500,000 passengers every workday. You may have to "join" the NYT website to read it, but it's free.
http://query.nytimes.com/gst/fullpage.html?res=9D0CE1DA1338F930A3575AC0A967958260&sec=&spon=&pagewanted=all
But I have also read that transit officials in Tokyo claim that city's Toei Mita line carries the most passengers in the world.
I realize this doesn't fully answer the question, but I hope it's a start. Compl. answer : RER A subway / interregional train line in Paris transports more than 1 000 000 people every work day, and 286 millions of people per year. http://en.wikipedia.org/wiki/RER_A
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The busiest New York City subway line has to be the 4-5-6 (the green line), as it is currently the only subway line that runs all the way up and down (north/south and south/north) Manhattan on the East Side.
A new East Side subway line, the 2nd Avenue line, is currently being built to ease the crowding on the 4-5-6 line. The first segment of the line is due to open sometime around 2013.