For a real algebraic variety, is the integral of the product of the Chern classes of two line bundles equal to the intersection number of the two corresponding divisors?
I'm not sure what you mean by a line bundle on a real algebraic variety and its Chern classes, but for smooth complex analytic manifolds the Chern class of a line bundle corresponding to a divisor is Poincare dual to the homological class of the divisor, as explained e.g. in GriffithsHarris, Chapter 1, Chern classes of line bundles. So the $c_2([D_1]\oplus [D_2])=c_1[D_1]c_1[D_2]$ is indeed the indeed Poincare dual to the intersection class of $D_1$ and $D_2$. If the manifold is a surface, then we get the a number. Maybe this not what you meant, but in that case you should really add some more details to your question. 

