The Compact Linear Collider (CLIC) is a proposed TeV-scale high-luminosity electron-positron collider at CERN. The first CLIC running stage, at 380 GeV, will focus on precision Higgs boson and top quark studies while the main aim of the subsequent high-energy stages, at 1.5 TeV and 3 TeV, is to extend the sensitivity of CLIC to different Beyond the Standard Model (BSM) scenarios. We studied the prospects for measuring invisible Higgs boson and additional heavy scalar decays using CLIC data at 380 GeV and 1.5 TeV. The analysis is based on the WHIZARD event generator, with fast simulation of the CLIC detector response parametrised by the DELPHES package. We present the expected limits for the invisible decays of the 125 GeV Higgs boson, the cross section limits for production of an additional neutral Higgs scalar, assuming its invisible decays, and limits on the mixing angle between the SM-like Higgs boson and the new scalar of the "dark sector" in the framework of the vector-fermion dark matter model.