Do Catholics object to autopsy? When a Catholic dies and there are questions as to cause of death do they object to a coroners order of an autopsy?
If yes what reasons do they give?
The only Catholic not permitted to be autopsied is the pope, due to Vatican protocol. 

At any time, a family may object to an autopsy of one of their family members, but this could be due to some a personally family reasons (perhaps a cultural one).

The Catholic Church teaches that both organ transplants and autopsies are morally acceptable in our times. 

For organ transplants the Catechism of the Catholic Church has the following to say:  


  Organ transplants are in conformity with the moral law if the physical and psychological dangers and risks to the donor are proportionate to the good sought for the recipient. Organ donation after death is a noble and meritorious act and is to be encouraged as a expression of generous solidarity. It is not morally acceptable if the donor or his proxy has not given explicit consent. Moreover, it is not morally admissible to bring about the disabling mutilation or death of a human being, even in order to delay the death of other persons. - Catechism of the Catholic Church, No. 2296.


For autopsies the Catechism of the Catholic Church has the following to say: 


  Autopsies can be morally permitted for legal inquests or scientific research. The free gift of organs after death is legitimate and can be meritorious. The Church permits cremation, provided that it does not demonstrate a denial of faith in the resurrection of the body. - Catechism of the Catholic Church, No. 2301.


Although popes may be embalmed after death, Vatican tradition does not permit the body of a pope to be autopsied. No pope has ever given this decision making ability to any Cardinal or to the College of Cardinals as a whole. 


  The current regulations regarding a papal interregnum were promulgated by John Paul II in his 1996 document Universi Dominici Gregis. During the vacancy caused by a pope's death the College of Cardinals is collectively responsible for the government of the Catholic Church and of the Vatican itself, under the direction of the Cardinal Chamberlain. Canon law specifically forbids the cardinals from introducing any innovation in the government of the Church during the vacancy of the Holy See.
  
  A dead pope's body then lies in state for a number of days before being interred in the crypt of a leading church or cathedral. The popes of the twentieth century were all interred in St. Peter's Basilica. A nine-day period of mourning (novem dialis) follows after the interment of the late pope. Vatican tradition holds that no autopsy is to be performed on the body of a dead pope. - Papacy


This tradition, however, would probably be waived if there were actual suspicion of serious foul play or something like that.