While you're on that planet, you are also in the same orbit around the star as the planet is. As long as you're in the same orbit as your vehicle (planet, spacecraft), you don't experience any acceleration relative to the vehicle. Whether you're on the planet, or you're just tagging along on the same orbit, is the same thing.
Similar question: in a satellite in a very eccentric orbit around the Earth, is any break in microgravity experienced in various parts of the orbit? Same answer - no. The satellite and everything in it follow the same orbit, so microgravity conditions are maintained.
Now things would be different if you had a very eccentric orbit around a source of much more intense gravity, like a neutron star or a black hole. At the lowest point in the orbit, you could experience tidal forces: the things closer to center experience stronger gravity compared to things further away from center.
But for this to be experienced within a small spacecraft, or within the limits of one human body, the gradient of the field would have to be very strong indeed; the orbit would have to reach very low, very close to the neutron star or black hole.
However, what you would experience would not be orbital "acceleration", but just tidal force. Even so, in extreme cases the tidal force could be strong enough to destroy the ship or kill the passengers.
See the short story "Neutron Star" by Larry Niven for a literary depiction (fairly accurate physically) of this phenomenon. Unfortunately, the story is spoiled by the explanation above, but hopefully learning about gravity compensates for spoiling the story. :)