The layered material ZrSiTe is currently extensively investigated as a nodal-line semimetal with Dirac-like band crossings protected by nonsymmorphic symmetry close to the Fermi energy. A recent infrared spectroscopy study on ZrSiTe under external pressure found anomalies in the optical response, providing hints for pressure-induced phase transitions at $approx$4.1 and $approx$6.5 GPa. By pressure-dependent Raman spectroscopy and x-ray diffraction measurements combined with electronic band structure calculations we find indications for two pressure-induced Lifshitz transitions with major changes in the Fermi surface topology in the absence of lattice symmetry changes. These electronic phase transitions can be attributed to the enhanced interlayer interaction induced by external pressure. Our findings demonstrate the crucial role of the interlayer distance for the electronic properties of layered van der Waals topological materials.