Depends on what you mean by "chain restaurant".

In North America at least, a case can be made for Delmonico's in New York City. They opened their first restaurant in Manhattan in 1827, and from 1837 to 1917, they operated at least two locations and as many as four locations simultaneously in different parts of Manhattan all under the Delmonico's name. They were back down to a single location at the end of 1917, and then their remaining location closed in 1923. A new Delmonico's under new management, but located at their most famous location (2 William Street) and retaining the original restaurant's menu, opened four years later in 1927, and continues to operate at that location today.

If you're thinking of the modern fast food chain restaurant, then it's almost certainly White Castle. They opened their first location in Witchita, Kansas, in 1921, and by the following year, they were operating nine hamburger stands in the city and two more locations in nearby El Dorado.

Sources:

"Delmonico's Hard Hit by the War", Chanute (Kansas) Daily Tribune, Oct 12, 1918, p.5

A History of Old New York Life and the House of the Delmonicos

Delmonico's, Fifth Avenue, New York

"For News Boys: White Castle Company Giving Hamburger and Pie to Each", Witchita Daily Eagle, Nov 29, 1922, p.9

EDIT: Fixed link.