The temperature and electric-field dependence of the electron emission from charged semiconductor quantum dots is studied with transient capacitance spectroscopy. The self-assembled InAs quantum dots are embedded within Schottky diodes grown with molecular-beam epitaxy on GaAs(001). In accordance with the different activation energies the emission from the s and the p shell of the quantum dots takes place with strongly different rates. In addition, the emission rates depend on the charge state of the shells. The behavior can quantitatively be understood with a thermionic-tunneling model in which the tunnel barrier is assumed to consist of a Coulomb barrier arising from the charge within the dot and a triangular contribution from remote charges.