  We study the collective modes of a low-energy continuum model of the
quasi-two-dimensional electron liquid in a layer of the organic compound
$\alpha-$(BEDT-TTF)$_2$I$_3$ in a perpendicular magnetic field. As testified by
zero magnetic field transport experiments and \textit{ab initio} theory, this
material hosts both massless and massive low-energy carriers, the former being
described by tilted and anisotropic Dirac cones. The polarizability of these
cones is anisotropic, and two sets of magnetoplasmon modes occur between any
two cyclotron resonances. We show that the tilt of the cones causes a unique
intervalley damping effect: the upper hybrid mode of one cone is damped by the
particle-hole continuum of the other cone in generic directions. We analyse how
the presence of massive carriers affects the response of the system, and
demonstrate how doping can tune $\alpha-$(BEDT-TTF)$_2$I$_3$ between regimes of
isotropic and anisotropic screening.
