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The class of warped product metrics can often be interpreted as key space models for the general theory of relativity and theory of space–time. In this paper, we study several non-Riemannian quantities in Finsler geometry. These non-Riemannian quantities play an important role in understanding the geometric properties of Finsler metrics. In particular, we find differential equations of Finsler warped product metrics with vanishing <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>χ</mi></semantics></math></inline-formula>-curvature or vanishing <i>H</i>-curvature. Furthermore, we show that, for Finsler warped product metrics, the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>χ</mi></semantics></math></inline-formula>-curvature vanishes if and only if the <i>H</i>-curvature vanishes.