Archaeologists always try to treat human remains with respect, even when they are so old that we don’t know their religion or their beliefs about death. Not all archaeologists feel that Richard III should be reinterred, because future developments in archaeological analytical techniques could eventually provide us with more information about this important find.
In the case of more recent excavated burials where we know the religion of the person, we try to treat their remains in accordance with their religious beliefs. In the case of Christian burials, the Church of England and English Heritage have agreed a set of guidelines for good practice, including an option for re-interment.
The Franciscan friars who conducted the burial would certainly have buried him with a proper, although perhaps minimal, funeral service using the Catholic rite of the time. Without this he couldn’t have been buried in the consecrated ground within the church building.
Richard III has already been buried with a Christian funeral so the re-interment will not be a funeral service. Instead it will be a Christian service celebrating his life with partners from other faiths, including Roman Catholic. It will include the multi-faith and multi-cultural communities of Leicester and modern-day England.
Richard III was king of England at a time before the Reformation, when the separation of what we now call the Church of England from what we now call the Roman Catholic Church took place. Neither of those names would have meant anything in his time. The concept of consecrated ground for burial is identical in the Roman Catholic Church and the Church of England. All Church of England churches founded before 1537 were originally what we now call ‘Roman Catholic’, and are full of burials that were originally Roman Catholic. So, the fact that Richard III was Roman Catholic should make no difference for his reburial in a beautiful medieval church building that is now a Church of England cathedral.
According to the English Heritage/Church of England guidelines [http://www.english-heritage.org.uk/publications/human-remains-excavated-from-christian-burial-grounds-in-england/], the relatives who need to be consulted are those who were close to the person and likely to have known them personally. Hence the accepted rule for burials over about 100 years old is that it is inappropriate to consult living relatives since it is unlikely that any of them will have known the deceased person personally. Archaeologists are, however, encouraged to be flexible around this 100 year threshold.
Richard III left no direct descendants. There are a great many distant relatives of Richard III’s immediate family alive today, scattered all over the world. Since Richard III lived so long ago, none of these are close relatives. So in the case of Richard III, the only surviving relatives are very distantly related, none will have known him personally, and there are far too many (running into millions) to trace and consult all of them.