The Church of Jesus Christ of Latter-day Saints is one denomination that teaches that God sorrows and weeps for those who choose to turn away from Him.
In 1830, Joseph Smith began work on what became known as the Joseph Smith Translation of the Bible. The most extensive part of this work is canonized in the Latter-day Saint standard works as “Selections from the Book of Moses” in the Pearl of Great Price. “Selections from the Book of Moses” is relatively short (8 chapters), but contains several poignant passages that are meaningful to Latter-day Saints, with a vision given to Moses – an illustration of the glory of God, the Creation, the Fall, and the Atonement of Jesus Christ, and the work of Adam and Eve, Enoch, and Noah in preaching faith and repentance. The entire work can be read at ChurchofJesusChrist.org.
Relevant to the question, in Moses chapter 7, Enoch prays and is called up to a mountain where he sees a vision. Starting in verse 28:
28 And it came to pass that the God of heaven looked upon the residue of the people, and he wept; and Enoch bore record of it, saying: How is it that the heavens weep, and shed forth their tears as the rain upon the mountains?
29 And Enoch said unto the Lord: How is it that thou canst weep, seeing thou art holy, and from all eternity to all eternity?
30 And were it possible that man could number the particles of the earth, yea, millions of earths like this, it would not be a beginning to the number of thy creations; and thy curtains are stretched out still; and yet thou art there, and thy bosom is there; and also thou art just; thou art merciful and kind forever;
31 And thou hast taken Zion to thine own bosom, from all thy creations, from all eternity to all eternity; and naught but peace, justice, and truth is the habitation of thy throne; and mercy shall go before thy face and have no end; how is it thou canst weep?
The Lord replied:
32 The Lord said unto Enoch: Behold these thy brethren; they are the workmanship of mine own hands, and I gave unto them their knowledge, in the day I created them; and in the Garden of Eden, gave I unto man his agency;
33 And unto thy brethren have I said, and also given commandment, that they should love one another, and that they should choose me, their Father; but behold, they are without affection, and they hate their own blood;
34 And the fire of mine indignation is kindled against them; and in my hot displeasure will I send in the floods upon them, for my fierce anger is kindled against them.
35 Behold, I am God; Man of Holiness is my name; Man of Counsel is my name; and Endless and Eternal is my name, also.
36 Wherefore, I can stretch forth mine hands and hold all the creations which I have made; and mine eye can pierce them also, and among all the workmanship of mine hands there has not been so great wickedness as among thy brethren.
37 But behold, their sins shall be upon the heads of their fathers; Satan shall be their father, and misery shall be their doom; and the whole heavens shall weep over them, even all the workmanship of mine hands; wherefore should not the heavens weep, seeing these shall suffer?
38 But behold, these which thine eyes are upon shall perish in the floods; and behold, I will shut them up; a prison have I prepared for them.
39 And that which I have chosen hath pled before my face. Wherefore, he suffereth for their sins; inasmuch as they will repent in the day that my Chosen shall return unto me, and until that day they shall be in torment;
40 Wherefore, for this shall the heavens weep, yea, and all the workmanship of mine hands.
41 And it came to pass that the Lord spake unto Enoch, and told Enoch all the doings of the children of men; wherefore Enoch knew, and looked upon their wickedness, and their misery, and wept and stretched forth his arms, and his heart swelled wide as eternity; and his bowels yearned; and all eternity shook.