With respect to negations, οὐ is typically used with a verb conjugated in the future tense, indicative mood, but μή is instead used with a verb conjugated in the aorist tense, subjunctive mood. Despite this, the meaning of both combinations can be translated and understood the same way.
Compare the following examples, both of which are translated into English as the well-known phrase, “You shall not murder”:
ΙΘʹ τὰς ἐντολὰς οἶδας Μὴ μοιχεύσῃς Μὴ φονεύσῃς Μὴ κλέψῃς Μὴ ψευδομαρτυρήσῃς Μὴ ἀποστερήσῃς Τίμα τὸν πατέρα σου καὶ τὴν μητέρα TR, 1550
ΙΗʹ λέγει αὐτῷ Ποίας ὁ δὲ Ἰησοῦς εἶπεν Τὸ Οὐ φονεύσεις Οὐ μοιχεύσεις Οὐ κλέψεις Οὐ ψευδομαρτυρήσεις TR, 1550
Regarding μή, Thayer wrote,3
as in the more elegant Grk. writ. where future things are forbidden (cf. Herm. ad Vig. p. 807), with the 2 pers. of the aorist subjunctive: μὴ δόξητε, Mt. 3:9; 5:17; μὴ φοβηθῇς, Mt. 1:20; 10:26, 31 [here L T Tr WH pros. impv. φοβεῖσθε], (alternating with the impv. pres. φοβεῖσθε in Mt. 10:28 [G L T Tr]); μὴ ἅψῃ, Col. 2:21; μὴ ἀποστραφῇς, Mt. 5:42; μὴ κτήσησθε, Mt. 10:9; add, Mt. 6:2, 7, 13, 31; Mk. 5:7; 10:19; Lk. 6:29; 8:28; 14:8; Jn. 3:7; Acts 7:60; Ro. 10:6; 1 Co. 16:11; 2 Co. 11:16; 2 Th. 2:3,—[in the last three exx. with the third pers., contrary to W. 502 (467)]; 1 Tim. 5:1; 2 Tim. 1:8; Rev. 6:6; 10:4 (μὴ γράψῃς, for ἔμελλον γράφειν precedes; but in Jn. 19:21 μὴ γράφε is used, because Pilate had already written); Rev. 11:2; 22:10, and very often. We have the impv. pres. and the aor. subj. together in Lk. 10:4; Acts 18:9.
While regarding οὐ, he wrote,4
As in Hebr. לֹא w. impf., so in bibl. Grk. οὐ w. 2 pers. fut. is used in emphatic prohibition (in prof. auth. it is milder; cf. W. § 43, 5 c.; also 501 sq. (467); [B. § 139, 64]; Fritzsche on Mt. p. 259 sq. [cf. p. 252 sq.] thinks otherwise, but not correctly): Mt. 6:5; and besides in the moral precepts of the O.T., Mt. 4:7; 19:18; Lk. 4:12; Acts 23:5; Ro. 7:7; 13:9.
Upon further examination, the LXX translation of the Hebrew text is a bit unusual. The initial Hebrew clause אַל יִתֵּן לַמּוֹט רַגְלֶךָ roughly translates as, “Let him not allow your foot to move.” The verb יִתֵּן is conjugated in the 3rd person (masculine gender, singular number), whose subject is presumably Yahveh.
On the other hand, the LXX translates the Hebrew into Greek as «μὴ δῷς εἰς σάλον τὸν πόδα σου», which may be roughly translated into English as “Do not allow your foot to move” where δῷς is conjugated in the 2nd person, rather than the 3rd person (like יִתֵּן), and the subject of δῷς is presumably the owner of the foot.
According to the LXX, the owner of the foot has the power to prevent his own foot from moving, while according to the Hebrew text, it is God who prevents the person’s foot from moving.
Wilke, Christian Gottlob. A Greek-English Lexicon of the New Testament: Being Grimm Wilke’s Clavis Novi Testamenti. Trans. Thayer, Joseph Henry. Ed. Grimm, Carl Ludwig Wilibald. Rev. ed. New York: American Book, 1889.
1 cp. Luke 18:20; Jam. 2:11
2 cp. Matt. 5:21; Rom. 13:9
3 p. 410, I, §6, d
4 p. 461, §6