With infrared spectroscopy we studied the bulk electronic properties of the topological antiferromagnet MnBi$_2$Te$_4$ with $T_N simeq 25~mathrm{K}$. With the support of band structure calculations, we assign the intra- and interband excitations and determine the band gap of $E_g approx$ 0.17 eV. We also obtain evidence for two types of conduction bands with light and very heavy carriers. The multiband free carrier response gives rise to an unusually strong increase of the combined plasma frequency, $omega_{mathrm{pl}}$, below 300 K. The band reconstruction below $T_N$, yields an additional increase of $omega_{mathrm{pl}}$ and a splitting of the transition between the two conduction bands by about 54 meV. Our study thus reveals a complex and strongly temperature dependent multi-band low-energy response that has important implications for the study of the surface states and device applications.
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