It is not a necessity, no.
Remember that even the Christian still struggles with sin & rebellion, even after he or she has been justified by the Cross of Christ. This is quite clear in Romans 7:13-25, where we resonate with Paul's daily struggle with his inability to live according to the righteousness to which he's been called. In that text, Paul paints the picture of the "inner self", or what we may also call "how the Father sees us" at war with our flesh, or the life that we've come out of. Paul mentions that the other members of his body (including our mind, passions, etc.) are still in that limbo state between doing the will of God and shedding our former sinful nature.
For the Christian, we are to test our desires, to see if they are of Christ and the Spirit, and then to pursue those which are with everything that we are. In essence, if we follow John 14:15, we'll be safe.
Love Jesus, do whatever you desire.
The catch is this: But in that order (love Jesus first, live it out second). Not every desire results out of loving Jesus.
(That is a paraphrasing of St. Augustine, but fairly accurate in my assessment).
To answer your original question for the person who is not yet a Christian: if the earthly fulfillment of their desire still leaves them longing (a la Plato's Forms in a sense, or more accurately a la Ecclesiastes 3:11), then it is most likely that it is indeed given to them by and for the Trinity.
In summary, it is not a necessity that a desire points to God, but the longing that results from an unfulfilled desire does point us to God. I would also caution against Platonic reasoning, because it's so antithetical to the Gospel. For more reading on that, check out NT Wright's Surprised By Hope. Fantastic explanation on how Plato and Western civilization clash with the Gospel and the Biblical understanding of "forms".
Hope this helps!