What is the ruling on one who does not act upon an authentic Hadith, denying it and claiming that the Prophet (peace and blessings of Allah be upon him) did not say it, and is he considered a disbeliever by doing so?

Whoever denies the Sunnah has disbelieved and exited Islam. As for whoever denies the authenticity of a specific sound hadith while affirming the Sunnah, he does not disbelieve. The hadith about killing an apostate is sound and authentic, and there is a consensus of scholars on it. So, whoever denies it should be informed, and if he persists, his matter should be referred to the judge.