It has been suggested that late-universe dark matter decays can alleviate the tension between measurements of $H_0$ in the local universe and its value inferred from cosmic microwave background fluctuations. Decaying dark matter can potentially account for this discrepancy as it reshuffles the energy density between matter and radiation and as a result allows dark energy to become dominant at earlier times. We show that the low multipoles amplitude of the cosmic microwave background anisotropy power spectrum severely constrains the feasibility of late-time decays as a solution to the $H_0$ tension.