Thoughts: Someone who worships other gods would open themselves up to the possession by these so-called-gods (if they existed- whom Moses and Paul called "demons"- Lev. 17:7; Deut 32:17; I Cor. 10:20,21). At least a portion of scripture recognizes a spiritual reality behind the worship of other gods.
So, Jesus, who- on the surface (and I believe in reality)- recognized the existence of the devil and demons did not want to help someone who may not be a believer, and who opened themselves- and their children- up to more infection by evil. He may have seen it like a person who puts a bandaid to heal a wound, only to have that the patient open the wound up again- making it worse instead of better. Jesus spoke of a concern to not heal someone of demons and then have that person open themselves up to more possession- which would be worse (Mt. 12:43-45). Yet, the woman came to him with a measure of faith, and a willingness to do whatever it takes (shown by her persistance) to heal her daughter. This contrasts with those who would say they can live unrepentantly like an unbeliever (especially in terms of sexual morality), and still hope to obtain blessing by Christ. Christ did not seek to bless her daughter despite the demon, but sought to bless the daughter by casting the demon out.
To me this indicates the Lord's willingness to help us, even if He knows we may fall back into more problems; even if we know we should not be in his line of focus; even if we feel God's power is limited. God hears our pain, and the heartfelt cry of a woman aching for her child. Jesus is not cold-hearted, or legalistic, but showing he knows how things ought to work, He goes beyond that to mercy- for God's blessings are not finite.
Prayer: Thank you, Lord, for your mercy and kindness.
John Calvin abridged: Jesus began to give a glimpse of his mercy to the Gentiles that would be freely offered after his death and resurrection. The woman was from a heathen nation- often the Jews would call all Gentiles Greeks. She had aquired some taste of piety for she called him "Son of David" (Mt. 15:22). Christ, at first, did not speak in order to whet her zeal and enhance her ardor. Often when God does not speak directly, He would have us remember his promises and character as Redeemer. It should take away our pride to note that we are reckoned "dogs" and to pour blessings upon us would be to "throw" them away. It also should astound us that we, who were once considered dogs, are now considered children (Eph. 2:14).