Rivnuts are permanent, blind fasteners used in many applications where a strong, reusable fastener is required. There are many sizes and styles of rivnuts, but the most common size for a 10-24 rivnut is 3/16.
When choosing a drill bit size, it is important to consider both the threads per inch (TPI) and the major diameter of the screw. The TPI will determine how fine or course the threads on the screw are, and this will affect how well the two mating parts fit together. The major diameter is simply the outside diameter of the screw (not including any head).
For example, a 10-24 thread has 24 threads per inch; this is considered a course thread. A 3/16 drill bit has a major diameter of 0.1875. This combination will result in a tight fit between the two parts with little play. If a finer thread were used (say, 12-24), then a smaller drill bit would be needed to create an equivalent fit.
In short, to answer your question - you should drill a 3/16 hole to install a 10-24 Rivnut.