Many sources, particularly Ashkenazi sources indicate that such a thing is possible. For example, the Sefer Hassidim (ed. Margolios 365) brings several examples of this from aggadic literature. For example, the Talmud (Berakhot 18b) recounts that the Ammora Sh'muel ascended to the heavenly residence and requested that the Ammora Levi to be admitted, and his request was accepted. Similarly, the Talmud (Sotah 7b) recounts that Moshe prayed that Yehuda be admitted into the heavenly academy, and his prayer too was accepted.
Additionally, the Talmud (Sotah 10b) states that David effectively prayed for the welfare of the deceased Avshalom.
Similarly, the Sifrei (Shoftim 210) states:
אומרים כפר לעמך ישראל. כשהוא אומר אשר פדית ה', מלמד שכפרה זו מכפרת על יוצאי מצרים. כפר לעמך, אלו החיים, אשר פדית, אלו המתים, מגיד שהמתים צריכים כפרה
That is, the elders of the city near which a murder victim is found, pray: 'forgive your nation Israel whom you redeemed'. 'Whom you redeemed' is interpreted as referring to the dead.
Similarly, The Pesikta Rabbati (ed. Ish Shalom Piska 20 Parashat Mattan Torah) states that prayers are effective on behalf of the dead:
שמא תאמר כיון שירד לגיהנם אין לו תעלה כיון שמבקשים עליו רחמים זורק אותו מגיהנם כחץ מן הקשת
[It should be noted that Buber writes that this is a later addition into the Midrash, see his comments in the introduction to Pesikta D'rav Kahanna 7].
This Midrash is quoted in various Ashkenazi sources, including Rokeah (217).
In a similar vein, there are various Ashkenazi sources that advocate performing mitsvot on behalf the deceased, such as Siddur Rashi (214).
However, this idea is not unanimously accepted. For example, R. Meir HaMeili (cited in Mesorah L'Yosef (Qafih) Vol. IV 2005, p. 105) encourages practices meant to benefit the dead, although they are difficult to reconcile with the idea that presumably people are judged upon death, and after that their status wont be changed, specifically since such practices will strengthen the fundamental belief in the afterlife among the masses.
Even more clearly, R. Avraham bar Hiyya writes (Hegyon HaNefesh HaAtsuvah p. 4) that any acts or prayers on behalf of the dead are useless:
כל החושב על מעשה בניו ובני עמו שהמ עושים בגללו אחרי מותוי
ומתפללים בעדו שהם מועילים לו מחשבת פתיות הוא ותוחלת שוא
Anyone who considers that the actions of his children or his people, that they perform for him after his death, and they pray on his behalf, that they are effective for him, it is a foolish thought, and false hope.
Similarly, the implication of Kohelet (9:5), (which R. Avraham bar Hiyya quotes), is that the after death one's fate is sealed:
כִּ֧י הַֽחַיִּ֛ים יוֹדְעִ֖ים שֶׁיָּמֻ֑תוּ וְהַמֵּתִ֞ים אֵינָ֧ם יוֹדְעִ֣ים מְא֗וּמָה וְאֵֽין־ע֤וֹד לָהֶם֙ שָׂכָ֔ר כִּ֥י נִשְׁכַּ֖ח זִכְרָֽם
For the living know that they will die, but the dead don't know anything, and they have no more reward, for their memory is forgotten.
Rashi explains (there):
אבל משמתו אינם יודעים מאומה ואין עוד להם שכר פעולה שיעשו מן המיתה ואילך אלא מי שטרח בערב שבת יאכל בשבת
Once they die they know naught, and they have no more reward for activities that they do from death on, rather, one who laboured on the eve of the Sabbath, will eat on Sabbath.
Although most directly he is addressing actions by the deceased; not others, the implication is that only one who earn reward in this world will receive it. After death, it is too late. (Cf. however Siddur Rashi cited above. One can of course debate the strength of this inference. In a responsum (V:49) Rashba cites this teaching as a problem for the view that one can improve the lot of the dead, but ultimately concludes that this teaching can be reconciled with improvement of the lot of the dead).
Similarly, Avot 4:17 states that one moment of repentance and good deeds in this world is better than the afterlife.
Rambam elaborates:
שאין אחר המוות שלימות ולא תוספת, ואמנם ישלם האדם ויוסיף מעלה בזה העולם...אלא במצב אשר ילך האדם - בו ישאר לעולם. ולזה ראוי להשתדל בזה הזמן הקצר המועט, ולא לאבדו אלא בקניית המעלות בלבד, לפי שאבדתו גדולה, הואיל ואין לו תמורה ואין לו תקנה.
For after death there is no completion or addition. Rather, a person completes himself and improves in this world...Rather, in the state that a person goes [to the afterlife], there he remains forever. Therefore, it is fitting for a person to exert himself in this short time, and not waste it with ought but self improvement, since there is a great waste in this, since there is no replacement, for this, and no possibility for rectification.
This is also the implication of Rav Sa'adya Gaon who writes in HaNivhar BeEmunot V'deot (Ma'amar 6 p. 204) that repentance is possible while a person is alive, but after death, there is no possibility to cleanse the soul for past deeds, and no hope for such as thing, Proverbs (11:7) states, when a wicked person dies, hope is lost:
כל זמן שהיא בגוף, אפשר לה לשוב להזדכך ולתנקות, ועל כן התשובה מקובלת בעוד האדם חי, וכאשר תצא ממנה אי אפשר לה להתנקות ממה שכבר יש בה, ואף אין לקוות לה מאומה מזה, כאשר אמר (משלי י"א ז') במות אדם רשע תאבד תקוה
Presumably they would reject or reinterpret the above Midrashim.
In this vein, regarding the Sifrei about the prayer recited with the heifer whose neck has been broken after a murder victim has been found, the Meiri (Temurah 15) brushes off the idea of atonement for the dead as a mere proposal in the midst of a discussion, for true forgiveness comes through repentance. Rambam (Hilkhot Rotseah Ushmirat HaNefesh 9:4) similarly reframes this as God forgiving the murder, rather than forgiving the dead, as noted by R. Eliyahu Nagar (Mesorah L'Yosef there p. 99).
Similarly, Maharam Halawa explains the Midrash about David praying for Absalom on the grounds that as the harmed party, David could effectively forgive Absalom for his actions. However, in general he maintains, (as we will quote below) that actions for the dead are ineffective.
Rav Hai Gaon rejects the idea that in general, one person's act can yield reward to a different party. This would seem to preclude performing practices to benefit the dead, but would not preclude the possibility that prayers for the dead could be effective. He writes in a responsum (Otsar HaGeonim Hagiga p. 28):
ואיך יעלה על לב כי שכרו של זה שעשה זה לזה, והלא הכתוב אומר צדקת הצדיק עליו תהיה וכן אמר רשעת הרשע עליו תהיה כשם שאין אדם נתפס בעון זולתו כך אין אדם זוכה בזכות זולתו
How could it occur to anyone that the reward for this that one person did would go to someone else? For the verse (Ezekiel 18:20) states: 'the righteousness of the righteous shall be accounted to him alone' and it also says (ibid): 'the wickedness of the wicked shall be accounted to him alone.' Just as a person cannot be caught by the sin of someone else, so too a person cannot receive merit from someone else's merit.
Maharam Halawa also rejects the idea that actions on behalf of the dead can have any benefit, and his explanation would seem to preclude the efficacy of prayer as well:
אין ספק בדבר כי לא יועיל ולא יציל מה שיעשה בשביל האדם אחרי
מותו כי כל אדם נדון לפי מה שהוא בעת מותו כי לפי מעלתו והשגתו
כשתפרד גופו מנפשו ישיג מעלה...ואין לה
תוספת מעלה ותועלת במה שיעשו אחרי כן להועילו ולהצילו מצרתו
There is no doubt that that which is done for a person after his death will not help him or save him, for a person is judged according to his status at the time of his death, for according to his greatness and achievement at the time of the separation of his body and soul, does he attain corresponding degrees of reward...and the soul gains nothing whatsoever from what they do after his death to help him or save him.
It should be noted that Maharam Halawa adduces proof for this position from the aforementioned Mishna in Avot, and Rambam's (aforementioned) explanation there.
Most of the above sources are culled from the 100+ page long article in M'sorah L'Yosef there.
