We have applied nuclear magnetic resonance spectroscopy to study the distinctive network of nodal lines in the Dirac semimetal ZrSiTe. The low-$T$ behavior is dominated by a symmetry-protected nodal line, with NMR providing a sensitive probe of the diamagnetic response of the associated carriers. A sharp low-$T$ minimum in NMR shift and $(T_1T)^{-1}$ provides a quantitative measure of the dispersionless, quasi-2D behavior of this nodal line. We also identify a van Hove singularity closely connected to this nodal line, and an associated $T$-induced Lifshitz transition. A disconnect in the NMR shift and line width at this temperature indicates the change in electronic behavior associated with this topological change. These features have an orientation-dependent behavior indicating a field-dependent scaling of the associated band energies.
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