If we are to think like the first-century hearers, we must recognize that the importance of the number fourteen is that it is a multiple of (that ever-so-important number) seven. Matthew is implying that the entire flow of God's history of creating a people for himself shows that Jesus the Christ is our Sabbath rest.
Forty-two, not Fourteen
Three sets of fourteen is six sets of seven. The operative number here is not really fourteen but forty-two. And no, forty-two is not the answer to life, the universe, and everything; actually it is the precursor to the one who is the meaning of everything. Jesus Christ is the seventh week of history; he is the completion and the culmination; he inaugurates the last days.
In the Lord Jesus, God's redemptive work comes to a rest. This does not mean that God becomes inert, but that that the finality and totality are invested in him, and history never moves beyond him, as he lives on in his indestructible life, ruling from on high.
Old Testament Prophecies
I am not aware of prophecies that reference the number fourteen. However, when this is seen as a matter of the seventh seven, all the significance of seven and its square come into play. There are a handful of references to forty-two in the Old Testament, but none of them shed a lot of light on this situation. The most relevant prophecy may be Daniel's reference to the week of weeks:
Now listen and understand! Seven sets of seven plus sixty-two sets of seven will pass from the time the command is given to rebuild Jerusalem until a ruler—the Anointed One—comes. Jerusalem will be rebuilt with streets and strong defenses, despite the perilous times. —Daniel 9:25 (NLT)
Admittedly, Daniel is giving a different perspective on the narrative than Matthew, but the same theological meaning of the seventh seven: the Messiah will be the fulfillment of all things.
New Testament Prophecies
The meaning of forty-two is further clarified by the last book of prophecy as the time of the dominion of the nations over God's people:
But do not measure the outer courtyard, for it has been turned over to the nations. They will trample the holy city for 42 months. —Revelation 11:2 (NLT)
Then the beast was allowed to speak great blasphemies against God. And he was given authority to do whatever he wanted for forty-two months. —Revelation 13:5 (NLT)
The Messiah is the one who brings this time to an end, defeating the enemies of God and rescuing his people. Upon his arrival, the time of trampling is over.
In Jon Ericson's answer, he links to an article by Bob Deffinbaugh, who points out the strong connections between Matthew 1 and Genesis 1. Indeed, what we see here is that just as God created progressively over six days, so God revealed himself progressively for six weeks, until on the seventh God himself appeared on earth and became our rest.