Is it proper in Judaism to pray for someone, even if they do not ask you to pray for them? Similarly, is asking someone to pray for you acceptable in Judaism? Usually, Christians ask each other to pray for one another, so I was not sure if this was also an accepted Jewish practice.
Asking someone to pray for you
Rabbi Moshe Feinstein writes (Igrot Moshe vol. 8 OCH ch. 43) that even according to those who forbid prayers to angels as intermediaries in any circumstance, utilising human intermediaries is permitted. This is true for prophets as evident from II Kings (4:23) and for Torah scholars as evident from Berakhot 34b.
אך איכא מחלוקת רבותינו הראשונים לבקש מהם שהם בשליחותם יעשו באופן היותר טוב לפנינו, דיש אוסרין גם זה. אבל לבקש אדם כשהוא בחיים להתפלל ולברך, הוא דבר הראוי והנכון.
Later in the piece he writes that utilising dead people as intermediaries, however, is more complicated. (1)
אך לאדם שמת, שכבר לא שייך שהוא יתפלל, אלא שנשמתו שהיא בגנזי מרומים שייך שתתפלל עבור אלו החיים, שלאלו דסוברים דלמלאכים אסור יש לעיין אם גם לבנ"א שמתו אסור
Whether or not R. Moshe's arguments are compelling, there is powerful precedent for his conclusion. In a letter to his brother-in-law(2), Rabbenu Avraham ben HaRambam writes:
Indeed we are in great danger I cannot relate to you [even] the fewest details, for I can imagine what your great consternation would be upon hearing them. Thus have I seen fit to omit them and to mention to you naught thereof. May our master not leave off praying for us, “since a captive cannot free himself” (Berakhot 5b). (Trans. Paul Fenton).
We see that Rabbenu Avraham himself requested that his brother-in-law pray for him. Thus he evidently holds that asking a human being to intercede on one’s behalf is permitted.
Praying without being asked
While it is hard to prove a negative, and therefore difficult to demonstrate that there is no problem, I am unaware of any reason why it would be a problem, and aware of several reasons and counterexamples that suggest that this act of kindness is fine.
We find many places in the Bible where people pray for others without being requested. For example, Abraham prays for the people of Sodom and Gomorrah (Genesis 18:23-32). Similarly, Moses prays for Miriam (Exodus 12:13)(3). Additionally, Moses prays on behalf of all of Israel, with no mention made of such a request (Exodus 32:30-2).
(1) It should be noted that it is possible that others could conceivably disagree with some of his proofs. For example, in the episode in Kings (v. 28; the cited verse seems to be a misprint) the Shunamite woman never explicitly asks Elisha to intercede with God. Rather, she merely says:
'Did I desire a son of my lord? did I not say: Do not deceive me?' (Trans. Mechon Mamre).
(2) Paul Fenton’s A Judeo-Arabic Commentary on the Haftarot by Hanan’el ben Semu’el (?), Abraham Maimonides’ Father-in-Law Appendix 1, p. 52 in Maimonidean Studies (ed. Arthur Hyman) vol. 1. New York, 1990.
(3) Although it could be argued that (12:11) was a request for prayer by Aaron for himself and Miriam, a) Miriam herself didn't ask him, and b) it appears that Aaron was just asking for forgiveness; not prayer. (cf. Rosh Moed Katan 3:7).
See Gemara Baba Kama 92a:
Rava said to Rabba bar Mari: From where is this matter derived whereby the Sages stated: Anyone who asks for compassion from Heaven on behalf of another, and he requires compassion from Heaven concerning that same matter, he is answered first? Rabba bar Mari said to him that the source for this is as it is written: “And the Lord changed the fortune of Job, when he prayed for his friends” (Job 42:10). Rava said to him: You said the proof from there, from a verse in the Writings, and I say the proof from here, from a verse in the Torah. As it is written: “And Abraham prayed to God; and God healed Abimelech, and his wife, and his maidservants, and they bore children” (Genesis 20:17), and it is written immediately following that: “And the Lord remembered Sarah, as He had said” (Genesis 21:1), with the pronoun interpreted homiletically: As Abraham said with regard to Abimelech. Because Abraham prayed for Abimelech that the women of his household should give birth, Abraham himself was answered concerning that matter.
See Orchot Tsadikim, chapter 9:
לכן כל איש אשר רצונו רצון האלהים – יצטער על שאין נעשה רצון השם. ואפילו על שונאו יתפלל אדם שיעבד לבורא יתברך. ויכוון בתפילתו בברכת "אתה חונן" ובברכת "השיבנו" ובברכת "סלח לנו" על כל ישראל, אוהביו ושונאיו, וכן בכל הברכות. כי איך יתכן שיתפלל "רופא חולי" ושאר הברכות, והוא לא חפץ שחברו יתרפא או יתחכם?! ועבור שזה היצר הוא רגיל מאוד בלבבות בני אדם, ואין מרגיש בעניין הזה, לכן נכתב להזהיר יראי השם להכין לבם לאלוקים בכוונה שלמה אמיתית, לשפוך נפשם נוכח פני השם על כל ישראל, אוהביו ושונאיו. ובזה יקיים "ואהבת לרעך כמוך" (ויקרא יט יח). "וטהר ידים – יסיף אומץ (איוב יז ט).
Every man holding the project of G-d, is saddened by people who don't make the G-d will. Even for his ennemy he have to pray that he make the G-d's will. in each blessing of the shmona Esre (אתה חונן, אשיבנו, סלח לנו) he have to include all Israel, friends and ennemies, ... (I have no English translation of the book).