Syntactically, "this longing is a part of God’s good gift, looking forward to the consummation" almost certainly means "this longing is [...] and looks forward to". In other words, the longing looks forward to the consummation in the sense of anticipating it, foreshadowing or prefiguring it, giving in advance a sense of what it will be like and directing attention to this future event.

I say almost certainly because it is possible to read "looking forward to the consummation" as explaining what the gift is: "God's good gift, namely looking forward to the consummation." That would mean that God's gift here is the pleasure the lovers get from looking forward to consummation. However, I think this is less likely to be the intended meaning. If it were, I think the writer would have put a colon after "gift" or said "God's good gift of looking forward [...]".

Of course, the lovers do in fact look forward to consummation in the sense of eagerly awaiting it, pleasurably anticipating it, but I don't think that's the meaning of "look forward to" here; at least, the syntax doesn't allow it to be read as "the longing is a part of God's gift and the lovers are looking forward to consummation".

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Note added at 1 hr (2016-09-27 11:08:44 GMT)
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Here are a couple of examples of this use of "look forward to" meaning "prefigure":

"Eat Drink Man Woman (EDMW) is a type of preview; it looks forward to Lee's later work and in this sense is a transitional film."
https://books.google.es/books?id=HPkIoOwlWtwC&pg=PA201&lpg=P...

"This assessment in the digression of Book V recapitulates earlier discussion and anticipates later. It looks back to the beginning of Book IV [...]. It looks forward to Books VI and VII, and to the metaphysical conception of the Form of the Good as what supplies the ultimate unifying explanation of everything there is."
https://books.google.es/books?id=UmQJZPvu6uYC&pg=PA217&lpg=P...