Three-body contributions to the van der Waals interaction potential for the ground state of the neon trimer are evaluated using the matrix Hartree-Fock model. Calculations are performed for two classes of nuclear geometries: (a) an equilateral triangle (D3h) with sides of 5·00, 5·25, 5·50 and 5·75 bohr; (b) an isosceles triangle (C2v) with two sides of 5·5 bohr and various angles between them. The site-site function counterpoise correction and the pairwise additive function counterpoise correction are used to investigate basis set superposition effects. The Axilrod-Teller triple-dipole term and higher-order non-additive dispersion terms are determined and compared with the non-additive matrix Hartree-Fock interaction energies for the range of nuclear geometries considered.