Yes. Psalm 22 is quite interesting in regards to the story.
Obviously, first and for most, Mathew 27:46, where jesus says:
"My God, My God, why hast thou forsaken me!" is a reference to the first line of psalm 22. This has been the source for the claim that this psalm is a prophecy of Jesus.
Interestingly, it has the following implications (not mutually exclusive)
1.) Psalm 22 clearly is written in the first person by a follower of god. Nowhere in the text of Psalm 22 does it suggest that the speaker is, in fact, god. But is merely one who is singled out and beaten upon by the wicked people surrounding him. The Psalm speaks of glory to come, regardless of the strife being observed today. That one day, the Lord's kingdom will be everywhere. that the "The meek shall eat and be satisfied: they shall praise the LORD that seek him: your heart shall live for ever." (inherit the earth, anyone?)
2.) it was a clearly intentional allusion in matthew. Whether or not this allusion was intentional by the author of matthew or by (in fact) jesus, is unkown. NO extrabiblical account exists of Jesus' words, so we can't know it is a fabrication. What we do know is that who ever made the allusion made it expressly with the idea that the speaker of the words (Why hast thou forsaken me) considered himself a man and not god.
So, if we do accept the idea that the psalm is prophecy, and prophecy fulfilled, christians must accept that this prophecy states Jesus was a man and not god.
But I do not believe that. It seems to me another example of the Jesus narrative being of redemption and the eminent coming of god. Remember that Christianity emerged from Jewish apocalyptic cults. Cults that would be familiar with passages like Psalm 22, which is incredibly apocalyptic in it's statements. The author of Matthew most likely made the allusion to emphasize the idea that god will win in the end and his followers will be the big winners, which is sort of the point of Psalm 22.
ETA:
Interestingly, a Jewish translation of the Jewish Psalm 22 states:
http://www.jewishvirtuallibrary.org/jsource/Bible/Psalms22.html
Psalm 22:17 " For dogs have encompassed me; a company of evil-doers have inclosed me; like a lion, they are at my hands and my feet."
Interesting how the bible translation has lacks the "like a lion" part.
The original Hebrew text states:
יז כִּי סְבָבוּנִי, כְּלָבִים: עֲדַת מְרֵעִים, הִקִּיפוּנִי; כָּאֲרִי, יָדַי וְרַגְלָי.
Notice the word:
כָּאֲרִי,
which translates as aryan, or like a lion
go one step further from:
http://www.behindthename.com/nmc/jew.php
ARI (1) אֲרִי m Hebrew
Means "lion" in Hebrew.
From another site we are given a bit more clarity of what this states:
"In Hebrew, the verse reads karah ari yad regal. Literally, mauled lion hands feet. Or, the lion mauled my hands and feet. The KJV, and virtually all Christian translations, completely ignores the word ari. It’s easy to see why they do that – because it significantly alters the context of the verse. If we’re talking about lions mauling the Psalmist rather than the Psalmist being pierced by some unknown entity, then it’s clear it doesn’t refer to Jesus. Fabricating messianic prophecy via incorrect translation is a modus operandi of Christians today and has been throughout history."
http://shemaantimissionary.tripod.com/id18.html
Do you not think it odd that the KJV translation of the bible completely omits this "like a lion" part?