An internal force is a force exerted by one part of a composite object on another part. By Newton's Third Law, if part $$A$$ exerts a force $$\vec F$$ on part $$B$$ then part $$B$$ will exert an equal and opposite force $$- \vec F$$ on part $$A$$. These internal forces can move parts $$A$$ and $$B$$ relative to one another, but when you consider the object $$A+B$$ as a whole (represented by a point mass at the centre of mass of $$A$$ and $$B$$) the internal forces cancel each other out. So, yes, internal forces always cancel out when you consider the motion of the centre of mass.
If, in addition to the $$\vec F$$, $$-\vec F$$ pair, part $$A$$ also exerts a force $$\vec G$$ on an external object $$C$$, then $$C$$ exerts a force $$-\vec G$$ on $$A$$. If we now consider the forces exerted on parts $$A$$ and $$B$$, we have $$\vec F$$, $$-\vec F$$ and $$-\vec G$$. The internal forces $$\vec F$$ and $$-\vec F$$ still cancel, but now there is an unbalanced force $$-\vec G$$ which will cause the centre of mass of $$A+B$$ to move.
If we extend our composite object to now include part $$C$$ as well, then we have four forces acting on parts of $$A+B+C$$; they are $$\vec F$$, $$-\vec F$$, $$\vec G$$ and $$-\vec G$$. Now $$\vec G$$ has become an internal force, and there is no net force to affect the motion of the centre of mass of $$A+B+C$$.