When a tribute dies, why does the hovercraft take the body away, and why does a cannon go off and why does the sky show the tributes and their deaths
The Capitol, is the primary ruling entity in the world of the Hunger Games. The Capitol, in contrast to other regimes seeks to remind the people that they, the Capitol are in charge and the lives of the people do not hold much meaning to them. The primary objective of conducting the Hunger Games is just 'muscle flexing' by the Capitol along with enforcing their iron grip on the country as the method in which dissent is removed (by making young men and women fight each other to death) is to send a message that rebellion and revolt against the Capitol will not work.
A parallel can be made with Oceania in 1984 wherein O'Brien informs Winston Smith that the objective of the Government (Big Brother) is not to create a utopia, but to create a world where there will be more pain, more fear and more oppression. They want to remove all dissenting thought against Big Brother and reserve happiness and laughter to the death and destruction of the enemy. O'Brien criticised the Communist and Fascist regimes of the past stating that they were not true to their ideology and did not dare to go the full way.
The action of taking the body away by the Hovercraft, denies the families of the tributes the opportunity to mourn them and give them a proper funeral. The sound of the cannon signifies the inevitable conclusion of the fight as one is expected to die in the Hunger Games and by sounding the blast, sends the message to the people and the families of the tributes about the deaths of their significant ones, adding in the salt as if it were into their wounds as they cannot even get an opportunity to properly mourn their dead as not even death can end the oppressive regime of the Capitol. The act of display of the names and deaths of the Tributes in the Sky is a formality, which shows to the people, how the game is going and betrays the true nature of the Capitol to the people.