The interaction between atoms and intense laser fields plays an important role in the field of ultrafast physics, which has become a powerful experimental technique to explore the structure and dynamics of matter. One of the critical issues is to explore the mechanism of laser-driven electron re-scattering process in, e.g., above-threshold ionization, high-order harmonic generation, and Non-Sequential Double/Multiple Ionization (NSDI/NSMI). Among them, the NSDI/NSMI is particularly interesting since it is a prototypical example for studying the electron-electron (e-e) correlation. During the past years, the general picture of NSDI has been established. In comparison, the NSMI is less well-understood although a series of experimental data have been collected, for example, extremely highly charged ions up to Ar¹⁶⁺, Kr¹⁹⁺, and Xe²⁶⁺ have been produced in super-strong laser fields. Here, the involvement of the multi-shell multi-electron, as well as highly nonlinear relativistic and non-dipole effects, makes the electron dynamics much more complicated. As a result, the theoretical explanation is hindered and lagged far behind partially due to the fact that solving the full-dimensional Time-Dependent Schrödinger Equation (TDSE) is currently limited to two-electron systems, leaving the dynamics of NSMI highly unexplored. On the other aspect, the Classical Trajectory Monte Carlo (CTMC) approach has been widely applied to atomic and molecular collisions and ionization of atoms by strong laser fields, which can be readily implemented in many-body systems and provide intuitive pictures to the dynamical processes of interest. It should be noted that, however, a classical multi-electron atom with Coulomb interactions is typically unstable and might suffer nonphysical auto-ionization, which needs feasible 'quantum' modifications. this article investigates the laser-driven lithium triple ionization by the classical trajectory Monte Carlo with the Heisenberg potential (CTMC-H) model. The model introduces the Heisenberg potential to mimic the Heisenberg uncertainty principle, which can be applied to obtain the classical stable configuration of the ground state of lithium atom by minimizing the system Hamiltonian to the values of the ionization energies. By solving the classical canonical equations of electrons, we study the total ionization rate of Li in a wide range of laser intensities. Due to the unique shell structure of the alkali metal elements, the out-most electron of the lithium atom is loosely bound and can be easily ionized, while the inner shell electrons are deeply bound and difficult to be deprived because of the much larger ionization energy. Therefore, we find that the single ionization is saturated over a wide span of laser intensities, and the double and triple ionization can be triggered by the re-collision of the out-most electron, showing the typical knee structure around 30 and 60 PW/cm², respectively. The difference between sequential triple ionization and non-sequential triple ionization is described in detail by plotting the momentum distribution of Li³⁺ in the polarization direction of the laser electric field, which shows a single peak structure near zero for the STI process and a double-hump structure in the NSTI regime as another signature of re-collision. Meanwhile, the magnetic effect is revealed by comparing the momentum spectra of Li³⁺ along the magnetic field polarization direction and the laser beam propagation direction. According to the energy distribution of three electrons mapped into the Dalitz diagram, we identify three types of re-collision mechanisms for the non-sequential triple ionization and reveal the fingerprint of thermalization induced by the (e, 3e) scattering mechanism. Besides, the electron re-scattering process is significant even in the sequential triple ionization region, e.g., at 100 PW/cm², as evidenced by the high-energy electrons located at the corners of the Dalitz plot. These findings not only provide deep insight into the mechanisms of the multiple ionization of alkali metal atoms in a femtosecond strong laser field but also have potential applications in manipulating the electron correlation on the attosecond time scale. Finally, we would like to emphasize that recently the magneto-optical trap recoil ion momentum spectroscopy has been proposed and established by combining cold atom trapping technology, strong laser pulse, and ultra-fast technology, exhibiting the ability to measure the full-dimensional momentum spectra of multiple reaction products and thus extending the study of strong-field multiple ionization and ultra-fast processes to the alkali metal atoms. We hope our theoretical predictions might stimulate experiments in this direction.