The solution of the variational problem which gives elastic strips in Minkowski 3-space is separately obtained for elastic strips with null and pseudo-null directrix. First, critical points of the modified Sadowsky functional (which depends on the modified torsion) for elastic strips with null directrix are characterized by three Euler-Lagrange equations. A connection is established between elastic curves on the two-dimensional null cone and elastic strips with null directrix. Then conservation laws of elastic strips with null directrix in Minkowski 3-space are given. Second, equilibrium equations for elastic strips with pseudo-null directrix are determined and solved. It is also shown the tangent and the binormal of a critical curve of the Sadowsky functional correspond to a null elastic curve in de Sitter 2-space and a spacelike elastic curve in the two-dimensional null cone, respectively. Finally, two conservation laws for elastic strips with pseudo-null directrix are derived.