An object is given an initial velocity. What will happen to the object if no other forces act on it?

It'll continue to be in motion.

Check Newton first law of motion

Newton's First Law states that an object will remain at rest or in uniform motion in a straight line unless acted upon by an external force

That means no Gravitational Force (g) acting on the object.

If we look the Newton's 2nd Law, we will see that the force (F) and gravitational force (g) is dependent each other, and the mass (m) is a coefficient between them.