Theoretically, if an object were to have negative rest mass, would it travel faster than light?
Which kind of mass are you talking about -- inertial mass, "active" gravitational mass, or the "passive" gravitational mass?
Generally, when considering negative masses, it is assumed that they fall with the same acceleration as as positive masses, and conservation of momentum is valid. They do not travel faster than light.
By putting '-m' in place of m in Newton's law of gravitation, you can play around a bit with the concept of negative mass. Note that negative masses move in the opposite direction of the applied force on it. Can you figure out what will happen when a mass m and a mass '-m' are just left close to one another?
To answer your question, objects with imaginary masses, which are called tachyons, travel faster than light.
EDIT: There is already a lot of discussion going on in https://www.physicsforums.com/showthread.php?t=66852, which I had somehow missed.
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