Which denominations would say that Jesus weeps for the souls in Hell?

Is it possible that Jesus is weeping right now whenever he sees souls going to Hell?

God does grant visions. And he grants these to whomever and whenever he wishes. After the death of the last Apostle John, these type of visions do not belong to the deposit of faith and therefore are not part of ALL that a believer ought to believe in order to be saved. The Church can and does weigh in on them for the good of souls, i.e. state whether or not the visions are of supernatural nature/origin; whether they are contrary to the faith, etc.

In some visions, as in the one reported in the OP, Jesus [and Mary] weep for souls. Here is an example:

Victim Soul for the Conversion of Communists
Jesus had chosen Edvige to be a victim soul and she offered herself generously for the salvation of others. Jesus asked her for many prayers and sacrifices because many souls were in danger of eternal damnation. Edvige wrote in her diary on May 25, 1941: “While I was praying before the Blessed Sacrament, I went into ecstasy and I saw Jesus on the Cross bleeding from each wound. The Blood was falling on the ground. I saw angels holding gold chalices up to the Wounds. The chalices filled up quickly and soon the Blood was falling on the ground again. Jesus was crying. I told Him, “Why are You crying?” He answered, “Daughter, I cry because so much of the Blood that I shed during My Passion is wasted without bearing fruit.” - Please see more at: The Servant of God, Edvige Carboni, (1880-1952) Mystic, Stigmatic, Victim Soul & Laywoman | Mystics of the Church

Can we find any hint from the Scripture?

• Heb 5:7: [H]e offered up prayers and supplications, with loud cries and tears[.]
• Heb 6:6: crucifying the LORD [again].
• Rev 2-3: Christ messages to the churches shows he is concerned about the lukewarmness of one, to another, he is urging repentance, etc.
• The ones in fredsbend's answer.

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.

The Scripture does tell us that Jesus can be sorrowful enough to cry:

When Jesus saw her weeping, and the Jews who had come along with her also weeping, he was deeply moved in spirit and troubled. 34 “Where have you laid him?” he asked.

“Come and see, Lord,” they replied.

Jesus wept.
John 11:33-35

Jesus was on his way to see Lazarus. He had just discovered that he had died and all of his family was there weeping when Jesus arrived. Jesus was moved, as any normal person would be, and wept as well. You probably missed this because it is the shortest verse in the Bible. ;)

Jesus certainly has the emotional capacity to cry, but does He still cry concerning the souls in Hell (or will he depending on belief)?

We know the Lord wishes that all would be saved:

God our Savior ... wants all people to be saved and to come to a knowledge of the truth.
1 Timothy 2:4

The Lord is not slow in keeping his promise, as some understand slowness. Instead he is patient with you, not wanting anyone to perish, but everyone to come to repentance.
2 Peter 3:9

But in my opinion, this doesn't mean He cries about it when He thinks about those who are not.

However, the image of Jesus that Christianity paints is of a man with deep love for His children. It is hard to reconcile that Jesus would not cry over the souls in hell in light of the compassion and love that Christianity says He has.

We can say assuredly that not Jesus, nor anyone, will weep in the New Heaven, Earth, and Jerusalem for any sorrowful reason.

Then I saw “a new heaven and a new earth,” for the first heaven and the first earth had passed away, and there was no longer any sea. I saw the Holy City, the new Jerusalem, coming down out of heaven from God ... God himself will be with them and be their God. ‘He will wipe every tear from their eyes. There will be no more death’[cf. Isaiah 25:8] or mourning or crying or pain, for the old order of things has passed away.”
Revelation 21:1-4

So it certainly is possible that Jesus weeps now for the lost, and even makes sense given Jesus' loving nature and certain Scriptures provided above, but we cannot Biblically argue that He will always weep for them, for in the End, there will be no mourning, crying, or pain.

As for denominations that teach this, I think this is one of those items left to the one preaching, whether they want to bring it up or not. I'm sure some catechisms and confessions of faith mention that God wants none to be lost, but I don't think they do or would mention that losing any makes Jesus cry.