Why does a torus not have points with zero curvature? By curvature I mean intrinsic curvature, if there even is such a thing as extrinsic curvature of a torus.
If a torus has negative curvature in some points and positive in others shouldn't there be points with zero curvature since it is a continuous surface?
 A: If by intrinsic curvature you mean Gaussian curvature, then a torus has points with zero Gaussian curvature. Namely the points that are "at the top" or "the bottom" of the torus when the revolution axis is vertical.
A: Your argument using continuity is essentially correct, as long as you augment it with the Gauss-Bonnet argument mentioned in the comment of @MichaelAlbanese. That argument tells you that the integral of sectional curvature with respect to area is equal to zero (and "sectional curvature" is what you ought to take as "intrinsic curvature"). 
As a consequence, either the sectional curvature is a constant equal to zero, or it has a nonzero value. Furthermore, if the sectional curvature has a nonzero value then it must have a value of opposite sign --- if, say, the value is positive at some point then the set of points where it has positive value has nonzero area, and the integral of sectional curvature over that set is positive, so there must exist points with negative sectional curvature so that the total integral of sectional curvature comes out to be zero. And finally, once one knows that the sectional curvature has both positive and negative values, your continuity argument applies.
