Hmm...despite what The Internet seems to claim in a lot of places, I'm not sure we can say Richard III was denied a Christian burial. Assuming he was indeed buried in the church at Greyfriars Priory, he was buried in consecrated ground. That implies Christian burial rites, and standing in full communion with Rome.

I looked up the Latin for Vergil (n.b.: not Virgil), who is usually cited for Richard's burial. It's "sine ullo funere honore," which is more like "without any funeral to honor him" or "without any funeral ceremony." This is a little different than being denied a Christian burial. The implication is that Richard still receives the actual requiem Mass/funeral rites. But it's a very stripped down version of normal procedure.

Similarly, just before this, the chronicles describe Henry having Richard's body on display for two days, basically just lying in a church covered in a sheet. The normal procedure would be for the deceased monarch to "lie in state," as it is called, in full honor. But here it's more a cursory glance at procedure as well as to demonstrate that Richard is well and truly dead.

The idea that Richard received an average but very quiet Christian funeral and burial seems to be backed up by the ad hoc nature of his initial burial, plus his re-burial with an actual memorial marker just a few years later. (But, uh, obviously not that much of a memorial marker, or...carpark, right.)

But you asked why, so obviously this cannot be the end of our story. ;)

Richard, of course, died a deposed and defeated king, and responsibility for his burial fell into the hands of his defeater and successor, Henry. Henry would have had one very important thing on his mind: preventing immediate revolts from any remaining supporters of Richard.

This is the pattern we see with the burial of an earlier deposed Richard, namely II. Henry IV was holding Richard in captivity, but when he got wind of a looming rebellion to restore Richard, somehow the ex-king ended up dead. (Somehow.) This Henry had that Richard buried in an unobtrusive convent rather than in prestigious, royal-affiliated Westminster Abbey where Richard had hoped and planned. There was no splashy funeral to galvanize and gather Richard's supporters, and no easy pilgrimage site or manufacture of relics out of his body.

Years later, with no longer any danger of revolt against Henry VI, he permitted Richard's body to be exhumed and finally buried alongside his wife as planned.

Skipping forward, it appears as though our Henry completed a parallel effort concerning Richard III. He buried the ex-king rather slapdash in a monastery without nearly the nation-wide reputation or royal connections/history like Westminster. But just a few years later, as mentioned above, he reburied Richard III in a marked, much more dignified (we think) tomb.

There's plenty of drama and story around Richard's fall and death. But his funeral and burial is kinda the opposite of that as far as historians are concerned--and being the opposite of that was exactly the story that Henry wanted.