Assuming that the mass of the planet is substantially lower than the mass of the star, then there won't be any change in the system's orbital dynamics (depending on the way the energy from the annihilated matter is distributed - see below). Unless the two masses are comparable, we can treat the planet as a test particle when studying its orbit and ignore its mass. The lowest mass stars are a bit under $0.07M_{\odot}$, or about 75 times the mass of Jupiter. The most massive planets (yes, the mass limit of a planet is a bit fuzzy) are generally around 13 Jupiter masses, so you would expect to see $M_p/M_*\lesssim1/6$. In this regime, Kepler's third law$^{\dagger}$ predicts that the planet will have perhaps a non-negligible influence on the orbits of both bodies. On the other hand, this is quite the edge case, and I'm not confident that you'd expect to see many ultra-high-mass planets orbiting ultra-low-mass stars. There are surely some, but not many.
Now, all of this assumes that the energy from the annihilated matter is released isotropically, the same in all directions. If not - for instance, if it's emitted in a beam of light in some direction - then conservation of momentum would mean that the planet will move in the opposite direction. But if we can assume that the emission is isotropic, like a star, there shouldn't be any effects on the orbit.
John Dallman makes a point that there would be issues with the internal structure of the planet - certainly true. Removing matter would constantly perturb it from from hydrostatic equilibrium. Assuming the process is gradual, the planet will then contract, reaching a new equilibrium when a chunk of mass is lost, then contracting into a new equilibrium when the next chunk is lost, and so on. I'd argue that in this sense, it's analogous to - though not the same as - a quasistatic process
The orbits of any moons around the planet would expand. Assuming that the moons retain their angular momentum (and I don't have any reason to say that they won't), the quantity $M_pa_m$ is conserved, with $a_m$ the semimajor axis of a moon. In other words, cut the mass of the planet in half and the semimajor axis of each moon will double. This is analogous to how Earth's orbit will expand as the Sun loses mass when our star enters the red giant, and, subsequently, asymptotic giant branch phases of its life.
$^{\dagger}$By the relation $a^3\propto(M_p+M_*)P^2$, which $a$ the semi-major axis and $p$ the orbit period. If $M_P\ll M_*$, we can approximate $M_P+M_*\approx M_*$, which is done if you want a quick-and-dirty model of a planet's orbit. If $M_P\approx M_*/6$, then I'd argue the planet's mass can no longer be ignored.
