The effect of Mn substitution, acting as a magnetic impurity for Fe, on the Dirac cone was investigated in Ba(Fe$_{1-x}$Mn$_x$As)$_2$. Both magnetoresistance and Hall resistivity studies clearly indicate that the cyclotron effective mass ($m^{ast}$) of the Dirac cone is anomalously enhanced at low temperatures by the impurity, although its evolution as a function of carrier number proceeds in a conventional manner at higher temperatures. Kondo-like band renormalization induced by the magnetic impurity scattering is suggested as an explanation for this, and the anomalous mass enhancement of the Dirac fermions is discussed.