My answer will pull together a number of posts on islam.se that are relevant here to shed some light on the issue; the TLDR is that I'm not aware of any resolution to the problem of hell in Islam. I will pull together the various concerns I'm aware of in this context as well as the attempts at a resolution and objections to those attempts.
The problem of hell is quite pronounced in Islam since the religion is very explicit on what merits eternal torture in hell: a lack of belief in the central tenets of Islam (see my answer here for some outliers). In particular, the problem of eternal hell (I already addressed this once here) is the more self-contradictory version of it.
Allah is called Ar-Rahman (the Most Merciful) and Al-'Adl (the Just; while some scholars do not consider this among the names of Allah, Justice is unanimously accepted as one of his attributes). It seems quite hard to track down anything reliable one what Allah's Mercy and Justice are (see my question here, which did not get any answers resolving the question). My ethical intuition - and that of many other people - tells me that "mercy" and "justice" in the human sense are not compatible with torturing someone for eternity. This is where the problem of hell arises.
As user TheRavenQueen points out in a comment to this other answer to this question, there is an inherent tension between mercy and justice:
Where to draw the line between mercy and justice? A just god should
not show mercy because it's not fair, a merciful god can't be just if
he shows mercy only to some.
Mercy has to do with forgiving and ignoring someone's offenses or missteps, justice is about treating people equally, correcting their offenses, and restoring an equitable situation. Vengeance is considered a part of that to some degree by most systems of ethics in the form of punishing someone to restore a feeling of dignity in the victim.
Allah is not just called merciful, he is called the Most Merciful, and it would in fact be disbelief to say anything in creation has more mercy than Allah. This immediately clashes with the ethical intuition of people who would not torture anyone for eternity - this is my ethical intuition, and in the first link above you will find that Ibn Arabi, the celebrated sufi scholar, said the same regarding his own.
Eternal torture in hell also clashes with many people's intuition of justice. A part of justice would be that the punishment has to be proportional to the crime (proportionality). There are at least the following reasons why this clashes with eternal hell:
The only offense for which Islam definitely threatens people with eternal torture in hell is disbelief (see e.g. 4:48). According to all theologians I'm aware of, they agree that anything less than disbelief only lands you in hell for a finite time. Historically, some sects disagreed here and said some sins that do not amount to disbelief will still land you in hell forever (e.g. some among the Mutazilites had such views as far as I know), but they were the exception and are considered kuffar otherwise.
Thus: the only offense that Islam considers worthy of eternal torture in hell is a lack of conviction in or dedication to its claims. If there was conclusive evidence that Islam is true, then the offense would be a lack of reasoning sufficiently, which hardly seems like an ethical failure.
Considering that there is no conclusive evidence that Islam is true (and as many will argue, some good reasons to suggest that it isn't), the situation is actually more striking since this means the offense is to not accept Islam because you see no reason to. Not accepting a claim as true because you don't have (sufficient, or any) evidence for it is not an ethical misstep however, it's the rational thing to do. I commented about claims from faith at length in the last section here.
Now comparing this to other offenses (genocide, mass-rape, warmongering, etc.), a lack of conviction in doubtful claims seems like the lesser ethical offense and should thus incur the lesser punishment, yet it does not according to Islam; so the injustice highlighted in this bullet point is a lack of consistency in the punishments of the hereafter according to Islam.
The next issue is that eternal torture is an unproportional punishment for an action that causes no one any harm. Since it is Allah you reject, if such an Allah exists, then that means it is Allah you offend. However, Islam is clear that Allah is free of all needs 4:131 and that disbelief cannot harm him 3:176. Thus the mental state of not being convinced of the claims of Islam harms no one, and yet the punishment is supposedly eternal torture. This is a crass lack of proportionality: the punishment is way too severe for the "crime".
Another issue with regard to proportionality is not about whether the punishment fits the "crime", but whether the punishment correlates appropriately with the accountability of the "offender". According to the Quran, Adam and Hawa (see e.g. 2:30 ff.) were created and lived in paradise, talked to angels, were commanded by Allah personally, i.e. had a host of convincing evidence that what Islam teaches is true. Iblis was in a similar situation before them. Iblis was sentenced to an eternity of torture for one instance of disobedience, Adam and Hawa are inhabitants of paradise despite their disobedience. And of course, Non-Muslims who are exposed to Islam now but see no convincing evidence for it at all are still considered on a path to eternal torture. So, someone disobeying despite having evidence that Islam is true could go to eternal torture or bliss, yet someone who has been presented the claims of Islam but finds that all the data marshalled in support of it is lacking and does not constitute evidence is destined for hell. In this sense, the punishment is unproportional to the culpability of the "offender".
This post gives a fair summary of the typical rhetoric found in scholarly literature; they don't see a problem of hell, they see Allah's eternal torturing disbelievers as a clear-cut case of his privilege as creator. The problem with this is that it relies on flawed analogies and ignores alternatives Allah has if he exists.
In particular, man's accountability is unclear in the face of the problem of free will and predestination which I asked a question about here: according to Islam, Allah creates every inclination, every bit of motivation, every act, every sensation, every thought a human has, every bit of environment that influences him. This makes analogies with human prison keepers and prisoners complaining about prison, or similar analogies, unapplicable since they talk about very different scenarios. The prison keeper did not create the prisoners and their criminal impulses. These analogies also fail because the prisoners are being punished for actually harming someone.
Assuming the free will vs qadr and justice issue is resolved such that man is accountable (and if we ignore that no one is being harmed by disbelief for a moment), Allah always has the option of simply unmaking the disbeliever, which would be much more merciful than torturing him forever. Alternatively, if someone objects to this by saying something like "well then Hitler never gets a just punishment" (replace "Hitler" with any arbitrarily violent person from history you like), then one can point out that Allah could create the world differently; there could be periods of testing followed by finite-time punishments, with a wheel of reincarnation that can eventually lead to paradise like in the vedic religions, or the scale of the possible harm could be limited to minor grievances by design, or both punishment and reward could be infinite in time by alternating between finite times of torture and bliss with the torture time spans diminishing over time (mathematically this is very easy to do), or a number of other options that an omniscient and omnipotent creator could realize. All of these settings seem to have less of a lack of proportionality than the claims about the afterlife that Islam makes.
In short, the problem of (eternal) hell has not received a lot of attention in Islamic discourse as far as I can tell, and the problems - while I would consider them to be significant, as would many Non-Muslims - are not considered as problems by most Muslim scholars. Notable high-ranking exceptions have been branded disbelievers by some for suggesting that hell must cease to exist (or at least to be torturous) after a finite time out of mercy or justice.