  TaIrTe$_4$ is an example of a candidate Weyl type-II semimetal with a minimal
possible number of Weyl nodes. Four nodes are reported to exist a single plane
in $k$-space. The existence of a conical dispersion linked to Weyl nodes has
yet to be shown experimentally. Here we use optical spectroscopy as a probe of
the band structure on a low-energy scale. Studying optical conductivity allows
us to probe intraband and interband transitions with zero momentum. In
TaIrTe$_4$, we observe a narrow Drude contribution and an interband
conductivity that may be consistent with a tilted linear band dispersion up to
40~meV. The interband conductivity allows us to establish the effective
parameters of the conical dispersion; effective velocity $v=1.1\cdot
10^{4}$~m/s and tilt $\gamma=0.37$. The transport data, Seebeck and Hall
coefficients, are qualitatively consistent with conical features in the band
structure. Quantitative disagreement may be linked to the multiband nature of
TaIrTe$_4$.
