There are two questions here, one quite answerable, the other less so.
What happens to Neo?
We actually see this scenario in the film's tie-in video game "The Matrix: Path of Neo". In short, Neo takes the blue pill and wakes up back in his apartment.
Since we know that the Agents are still aware of him, but lacking any further connection to the Zionese rebels, the Agents would probably put another bug in Neo's tummy (or his eye) and leave him in situ.
What happens to the Matrix?
There's no real indication of what happens if The One won't follow the Path of The One, but my personal instinct is that if they point-blank refuse to do so (or get their fool self killed before they get to the Architect), the Machine Mainframe would likely just place the Prime Program into another newborn. The machines can presumably wait a few extra years before destroying Zion if needs be.