Serre-Swan's theorem (see the MO discussion) says that any locally free sheaf over an affine variety is a direct summand of a free sheaf. However, this is not true on projective varieties. It is not hard to check that a non-trivial line bundle with non-zero global sections can not be a direct summand of a free sheaf. The reason is that, being a direct summand of a free sheaf implies that the dual line bundle also has non-zero global sections. But that implies the line bundle is trivial.

I am wondering if the same holds for vector bundles, i.e. if a vector bundle and its dual over a projective variety both have non-zero global sections, then the vector bundle is trivial.

Another I think related question is the following:

Is a direct summand of a direct sum of line bundles on a projective variety also a direct sum of line bundles?