Many Christians believe Psalm 22 refers to Jesus of Nazareth. How have Jews historically understood Psalm 22 and interpreted it?
This article, by Prof. Rivka Ulmer, might answer some of your questions... She writes (pg. 108): "Prior to the attestation in the New Testament, there is no evidence of Psalm 22 being used in a Jewish messianic context... Jewish interpretations of the Psalm identify the individual in the Psalm with a royal figure, alternatively interpreted as King David, King Hezekiah, or Queen Esther." She discusses several early Jewish and Christian sources, including the ones cited above:
Megillah 15b (her translation):
And stood in the inner court of the king’s house (Esther 5:1). R. Levi said: When she reached the chamber of the idols,the Divine Presence left her. She said, My God, My God, why have You forsaken me? (Ps. 22:2). Is it possible that You punish the inadvertent sin like the presumptuous one, or one done under compulsion like one committed willingly? Or is it because I called [Ahasuerus] “dog,” as it says Save my soul from the sword, my only one from the power of the dog? (Ps. 22:21). She immediately retracted and called him “lion,” as it says, Save me from the lion’s mouth (Ps. 22:22).
She discusses many others, including Midrash Tehillim and its interpretation of this psalm as referring to David's life as a shepherd (which is too lengthy to type out here but you can read it in Esther Menn's article here), as well as how this psalm is (mis)translated and utilized in the Christian tradition. I hope this helps.
A good place to look to find refutations of Christian messianic interpretations of the bible is Sefer Nitzachon, printed in Otzar Vikuchim by Dr. J. D. Eisenstein.
This is his answer to this specific case (p. 256): The Christian claim is that Jesus was crying to G-d, his father, "Why have you abandoned me?" at the time he was being executed. But according to Christian philosophy, Jesus wanted to go down to earth in flesh and blood in order to save all the sinners from hell through his death. If he really wanted to die in order to atone for mankind, why was he crying out why G-d abandoned him? Would he really say "I will call out but I am not answered" if he didn't want or intend to be answered? Furthermore, he refers to "our fathers" (v. 5) — but Christians claim he is the son of G-d, and if so how could he have multiple fathers? Would a representation of G-d on earth call himself a "worm" (v. 7; read תולעת for תועלת in the book)? How can he refer to his brothers (v. 23) if an only son of G-d has no brothers? Why does he praise the descendants of Ya'akov and Yisra'el for honoring G-d (v. 24) if they were the ones killing him, the son of G-d?
This blog post quotes the Radak as challenging the Christian interpretations in pretty much the same words. (This challenge is censored from most current editions of his commentary.) I would guess that the commentary of the Radak was used as a source by the author of Sefer Nitzachon.
To add just one Jewish interpretation of the psalm, the Targum (online here, apparently translated into English here) of an uncertain date reads it, like Rashi, as referring to the state of Israel among the nations. Other answers have already preceded this one, and better address Jewish interpretations of the Psalm, whereas this one focuses more on polemics against the Christian interpretation.
In an old Jews for Jesus brochure I saved from my college days, there is a section that quotes several Biblical verses which they say foretell the life of Christ. One of these is Psalms 22:16 (verse 17 in our Bible), which they translate as "They pierced my hands and feet." This supposedly foretells the crucifixion of Jesus where his hands and feet were pierced by the nails that hung him to the cross. One problem, it doesn't work in Hebrew.
The Psalm describes the angst of the psalmist who is surrounded by enemies and asks why G-d has forsaken him (Rabbi David Kimchi (the Radak b 1160) said this refers not to a person, but to the Jews in the Babylonian exile). Psalms 22:17 says in Hebrew: "k'ari b'yadai v'raglai" ("Like a lion [the enemies] are at my hands and feet"). The disputed word here is "k'ari" which is spelled kaph - aleph - resh - yud. Most graduates of a Hebrew school education know that an ari is a lion, and that the use of the letter "kaph" before a word means "like" or "as." The Christians appear to have invented a new Hebrew word which they pronounce "koari" yet no such word exists in Hebrew with the same spelling. There is a similar sounding word to koari that is used to mean to dig, or perhaps bore (as in a hole), although there are better words for that. But the spelling is much different. In "koari" there is no letter aleph as there is in the word k'ari and no grammatical reason for dropping it.
Judah Himango (a messianic Jew, aka Christian) wrote (in 2014) "This answer suggested Christian bibles mistranslate Psalm 22. However, the Dead Sea Scrolls, written before the time of Jesus, contain the very translation Christians use. See here: torahresource.com/EnglishArticles/Ps22.16.pdf." This is incorrect. The Dead Sea Scrolls do not contain the "very translation Christians use." They confirm the Jewish translation of "like a lion."
There are two fragments containing Psalms 22:17 in the Dead Sea Scrolls (DSS). 4QPs-f; known as the Qumran MS, does not have the word in question.
In the second fragment, found at Nahal Hever (HHev/Se 4 (Ps); known as the Bar Kochba MS, the word is preserved. Christians claim it is "pierced" -- and this is 100% incorrect.
The word claimed to be pierced by Christians in this psalm is כָּרוּ karu -- which actually translates to "they dug" -- and is not the word for "pierced." So "they dug my hands and feet"??
This verb is never used in the sense of "pierce"; it means to dig (excavate) in the ground. "Pierce" would be a completely different verb.
The word כָּרוּ karu ("they dug") occurs in T'hillim 57:7 and 119:85, as well as in Yirm'yahu 18:20 and 18:22, but nowhere in T'hillim, chapter 22. This form is past tense, 3rd person masculine singular: the full conjugation is
∙ כָּרִֽיתִי kariti "I dug" (B'réshιt 50:5) ∙ כָּרִֽיתָ karita "you [m.sing.] dug" (T'hillim 40:7) ∙ כָּרִית karit "you [f.sing.] dug" ∙ כָּרָה karah "he dug" (T'hillim 7:16, Divrei Hayamim Beit 16:14) ∙ כָּרְתָה kar'tah "she dug" ∙ כָּרִֽינוּ karinu "we dug" ∙ כְּרִיתֶם k'riy'tem "you [m.pl.] dug" ∙ כְּרִיתֶן k'riy'ten "you [f.pl.] dug" ∙ כָּרוּ karu "they dug"
But the Christians have an even bigger problem. The DSS scroll fragment containing this word does NOT have the word כָּרוּ karu ("they dug").
The word they claim actually does not exist in the Hebrew language. Those who claim the word 'karu' has an alef in it are being dishonest. There is no such word in the Hebrew language, nor is there a root of a verb that spells "kaf-alef-resh" in the Hebrew language.
The fragment HHev/Se 4 (Ps) shows the Hebrew letters (kaf), (aleph), (resh), and what appears to be a somewhat elongated letter (yod), which some perceive to be the letter (vav). Thus, the reading of this word would be either (ka'ari) or (ka'aru).
It cannot be ka'aru as many Christians suggest, because there is no root verb containing the letter (aleph) in it, conjugated in this fashion (3rd-person, plural masculine gender, past tense), with the meaning of they pierced -- as in most Christian translations, or even as "they dug."
Without the letter (aleph), and using, for the moment, the argument that the last letter [the elongated (yod)] is a (vav), the word would be (karu), for which the Hebrew root verb is (karah), [to] dig [in dirt], such as digging a ditch (e.g., Ps 57:7). In other words, (karu) has the meaning [they] dug [in dirt]. This verb is never used in the context of piercing, either literally or metaphorically, in any of its 15 applications in the Hebrew Bible.
The fragment has caused debate because it contains an elongated letter (yod) that resembles the letter (vav). Since the word (ka'aru) does not exist in the Hebrew language, the most plausible explanation is that such discrepancy in the fragment is simply a case of scribal variation (or error).
Rabbi Tovia Singer wrote an excellent article about this issue -- with images of the fragment. http://outreachjudaism.org/crucifixion-psalm/