We report transport measurements on GaSb-InAs-GaSb heterostructures under hydrostatic pressure (up to 1.2 GPa) and high magnetic field (up to 18 T) at low temperature. The pressure-induced decrease of the carrier concentrations is analyzed in terms of two distinct causes: an intrinsic charge transfer between the GaSb valence band and InAs conduction band, and an electron transfer to interface donor states. Using a simple self-consistent variational approach to model the structure and the experimental values of the concentration rates of decrease, we are able to estimate (1) the effective rate at which the band discontinuity \ensuremath{\Delta} at the interface decreases as we apply pressure (d\ensuremath{\Delta}/dP=67 meV/GPa) and (2) the density of interface states (${n}_{T}$\ensuremath{\approxeq}4\ifmmode\times\else\texttimes\fi{}${10}^{12}$ ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}2}$ ${\mathrm{eV}}^{\mathrm{\ensuremath{-}}1}$). Nonparabolicity of the InAs conduction band has been taken into account in the calculations.