Short answer
They are the inner sins and flaws that spoil faith and deeds: arrogance, envy, showing off, self-admiration, love of the world, malice, bad assumptions and hypocrisy. Hanafi scholars, with the other schools, hold that learning about them and working to cure them is an individual obligation, because Allah looks at what is in the heart.
Knowing the diseases of the heart, such as kibr (arrogance), hasad (envy), riya (showing off) and 'ujb (self-admiration), and striving to remove them, is fard 'ayn on every adult Muslim, as Ibn Abidin states. Acting on them is haram.
The four schools agree on this.
Explanation
The Quran speaks of hearts that are diseased (2:10) and says that on the Day of Judgement nothing will benefit a person except coming to Allah with a sound heart (26:89). The Prophet ﷺ said there is a piece of flesh in the body which, when sound, makes the whole body sound, and when corrupt, corrupts it: the heart. He also said that Allah does not look at your appearance or wealth, but at your hearts and deeds.
- Kibr (arrogance): rejecting the truth and looking down on people. The Prophet ﷺ said no one with an atom's weight of it in his heart will enter Paradise.
- Hasad (envy): wishing a blessing would leave someone else.
- Riya (showing off): doing good deeds to be seen.
- 'Ujb (self-admiration), and love of the world that pushes the Hereafter aside.
- Hiqd (malice) and su' al-zann (bad assumptions about people).
- Nifaq (hypocrisy) and doubt in faith, the disease the Quran mentions most.
These are moral faults, not medical conditions: depression and anxiety are not diseases of the heart in this sense. The classical cure, as taught by scholars such as al-Ghazali, is to recognise the fault in yourself and act against it: praise the one you envy, give in secret what you are tempted to show, serve the people you feel above. Keep dhikr, recite the Quran, keep good company and make dua.
The Prophet ﷺ often said: Ya muqallibal-qulub, thabbit qalbi ala dinik, O Turner of hearts, keep my heart firm on Your religion. The four schools agree on all of this. For daily remembrance that softens the heart, see the dhikr list.
The evidence
Quran, Surah Al-Baqara 2:10
فِى قُلُوبِهِم مَّرَضٌۭ فَزَادَهُمُ ٱللَّهُ مَرَضًۭا ۖ وَلَهُمْ عَذَابٌ أَلِيمٌۢ بِمَا كَانُوا۟ يَكْذِبُونَ ﴿١٠﴾
In their hearts is disease, so Allah has increased their disease; and for them is a painful punishment because they [habitually] used to lie.
The Quran describes hypocrisy and doubt as a disease of the heart.
Quran, Surah Ash-Shu'araa 26:88-89
يَوْمَ لَا يَنفَعُ مَالٌۭ وَلَا بَنُونَ ﴿٨٨﴾ إِلَّا مَنْ أَتَى ٱللَّهَ بِقَلْبٍۢ سَلِيمٍۢ ﴿٨٩﴾
The Day when there will not benefit [anyone] wealth or children But only one who comes to Allah with a sound heart."
A sound heart is what saves on the Day of Judgement.
Sahih al-Bukhari 52
Sahih... ألا وإن في الجسد مضغة إذا صلحت صلح الجسد كله ...
"In the body there is a piece of flesh: if it is sound, the whole body is sound."
The state of the heart governs a person's deeds.
Sahih Muslim 2564
Sahih... إن الله لا ينظر إلى صوركم وأموالكم ولكن ينظر إلى قلوبكم ...
"Allah does not look at your forms or your wealth, but He looks at your hearts and your deeds."
Allah judges what is in the heart.
Sahih Muslim 91
Sahih... لا يدخل الجنة من كان في قلبه مثقال ذرة من كبر ...
"No one who has an atom's weight of arrogance in his heart will enter Paradise."
Arrogance is among the gravest diseases of the heart.
Jami' at-Tirmidhi 2140
Grading to be checked... يا مقلب القلوب ثبت قلبي على دينك ...
"O Turner of hearts, keep my heart firm on Your religion."
His frequent dua for a steadfast heart.
Where the ruling is recorded
- Radd al-Muhtar (Ibn Abidin), Muqaddimah, on knowledge that is an individual obligation. Counts knowing sincerity, self-admiration, envy and showing off among what every Muslim must learn.
- Ihya Ulum al-Din (al-Ghazali), Rub' al-Muhlikat (the quarter on destructive vices). The classical treatment of each disease of the heart and its cure.
Checked by our team. For a more accurate answer to your own situation, consult a scholar. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.