We present a $^{125}\mathrm{Te}$ nuclear magnetic resonance (NMR) study in the three-dimensional spin web lattice ${\mathrm{Cu}}_{3}\mathrm{Te}{\mathrm{O}}_{6}$ which harbors topological magnons. The $^{125}\mathrm{Te}$ NMR spectra and the Knight-shift $\mathcal{K}$ as a function of temperature show a drastic change at ${T}_{\mathrm{S}}\ensuremath{\sim}40\phantom{\rule{0.28em}{0ex}}\mathrm{K}$ much lower than the N\'eel ordering temperature ${T}_{\mathrm{N}}\ensuremath{\sim}61\phantom{\rule{0.28em}{0ex}}\mathrm{K}$, providing evidence for the first-order structural phase transition within the magnetically ordered state. Most remarkably, the temperature dependence of the spin-lattice relaxation rate ${T}_{1}^{\ensuremath{-}1}$ unravels spin-gap-like magnetic excitations, which sharply sets in at ${T}^{*}\ensuremath{\sim}75\phantom{\rule{0.28em}{0ex}}\mathrm{K}$, the temperature well above ${T}_{\mathrm{N}}$. The spin-gap behavior may be understood by weakly dispersive optical magnon branches of high-energy spin excitations originating from the unique corner-sharing Cu hexagon spin-1/2 network with low coordination number.