Virtue of Forgiving Others in Islam
Question

What advice can you give to someone about the necessity of forgiving people and being patient with their annoyance, not only forgiving them outwardly but also inwardly, i.e., sincere forgiveness that comes from the heart, that is not tainted with any element of resentment towards anyone?
Praise be to Allah.Virtue of forgiving others in Islam
The Muslim is the brother of his fellow Muslim; he loves for him of good what he loves for himself, and he hates for him of evil what he hates for himself.
Undoubtedly there is no one among us who does not make mistakes and fall short, and does not wrong other people or transgress their rights to some extent; each of us would like people to forgive him for all the wrongs he has done to them, so that they will not ask for restitution thereof on the Day of Resurrection, which is the time when he will be most in need of his good deeds.
Forgiving others in the Quran
Allah, may He be exalted, urges us in His Book to forgive people and be patient with their annoyance, as He says (interpretation of the meaning):
“Who spend [in the cause of Allah ] during ease and hardship and who restrain anger and who pardon the people - and Allah loves the doers of good” [Al ‘Imran 3:134]
“If [instead] you show [some] good or conceal it or pardon an offense - indeed, Allah is ever Pardoning and Competent” [an-Nisa 4:149]
“And if you punish [an enemy, O believers], punish with an equivalent of that with which you were harmed. But if you are patient - it is better for those who are patient” [an-Nahl 16:126]
“And whoever is patient and forgives - indeed, that is of the matters [requiring] determination” [ash-Shura 42:43]
“But if you pardon and overlook and forgive - then indeed, Allah is Forgiving and Merciful” [at-Taghabun 64:14].
Hadiths about forgiving others
This is something that is also mentioned a great deal in the Sunnah. Muslim (4689) narrated from Abu Hurayrah that the Messenger of Allah (blessings and peace of Allah be upon him) said: “Charity does not decrease wealth. No one forgives, but Allah increases him in honour, and no one humbles himself before Allah but Allah raises him in status.”
Ahmad (21643) narrated that ‘Ubadah ibn as-Samit said: I heard the Messenger of Allah (blessings and peace of Allah be upon him) say: “There is no man who suffers an injury to his body, then he forgives [the one who caused it] as an act of charity, but Allah will absolve his sins commensurate with the extent of his charity.” (Classed as sahih by al-Albani in as-Sahihah, 2273)
Ahmad (1584) narrated that ‘Abd ar-Rahman ibn ‘Awf said: The Messenger of Allah (blessings and peace of Allah be upon him) said: “There are three things concerning which, by the One in Whose Hand is the soul of Muhammad, I would certainly swear an oath: wealth does not decrease because of charity, so give charity; no man pardons another for some mistreatment, seeking thereby the pleasure of Allah, but Allah will raise him in status thereby; and no one asks of people but Allah will increase him in poverty.” (Classed as sahih by al-Albani in Sahih at-Targhib, 2426)
And Ahmad (6255) narrated from ‘Abdullah ibn ‘Amr ibn al-‘As from the Prophet (blessings and peace of Allah be upon him) that he said, when he was on the minbar: “Show mercy, and you will be shown mercy; forgive, and Allah will forgive you.” (Classed as sahih by al-Albani in Sahih at-Targhib, 2465)
When Mistah ibn Uthathah indulged in gossip during the slander incident (al-ifk), and Allah sent down revelation confirming the innocence of ‘Aishah (may Allah be pleased with her), Abu Bakr, who had been spending on Mistah because he was his relative and was poor, said: By Allah, I shall never spend anything on Mistah again after what he said about ‘Aishah. Then Allah revealed the words (interpretation of the meaning):
“And let not those of virtue among you and wealth swear not to give [aid] to their relatives and the needy and the emigrants for the cause of Allah , and let them pardon and overlook. Would you not like that Allah should forgive you? And Allah is Forgiving and Merciful” [an-Nur 24:22].
Abu Bakr said: Yes indeed, by Allah, I would like Allah to forgive me. So he resumed spending on Mistah as he used to do, and he said: By Allah, I shall never deprive him of it.
This hadith was narrated by al-Bukhari (4381) and Muslim (4974).
Undoubtedly Allah does not bestow such a great characteristic upon anyone except the elite and best of His creation. The Muslim should take the righteous as his example. What could be more difficult than having a poor relative on whom you spend and whom you treat kindly, then he casts aspersions on your honour by speaking ill of it? Yet despite that, Abu Bakr resumed spending on Mistah (may Allah be pleased with them both) and swore that he would never stop doing so. This can only be due to his purity of heart and his love of doing good, pardoning and forgiving, otherwise he would not have sworn to never stop spending on him.
How to react to people's annoyance and mistreatment 
One of the best ways in which a Muslim can help himself to bear with patience whatever befalls him of annoyance and mistreatment from other people is to think of his own shortcomings in his duties towards Allah, and how much he would like Allah to pardon him and conceal his faults. This is how Allah, may He be exalted, encouraged Abu Bakr as-Siddiq to forgive Mistah, as He said: “and let them pardon and overlook. Would you not like that Allah should forgive you? And Allah is Forgiving and Merciful”. In other words, deal with people on the basis of pardoning and overlooking, as you would like them to treat you in the same manner, for the reward is of the same nature as the deed.
Hence Allah, may He be exalted, promised the one who would behave in this manner (forgiving people) a high status, for his reward is due from Allah. Allah, may He be exalted, says (interpretation of the meaning):
“but whoever pardons and makes reconciliation - his reward is [due] from Allah” [ash-Shura 42:40].
Shaykh as-Sa‘di ( may Allah have mercy on him) said: 
“Stating that the reward of the one who forgives is due from Allah motivates him to forgive and to treat others as he wants Allah to treat him. Just as he wants Allah to forgive him, he should forgive others, and just as he wants Allah to pardon him, he should pardon others, for the reward is of the same nature as the deed.” (Tafsir as-Sa‘di, p. 760)
How to free your heart of grudges and rancour
One of the greatest means that will help the Muslim to free his heart of grudges and rancour is to be keen to show sincerity towards the Muslims in general, and to do that for the sake of Allah, because he knows that Allah is pleased with that and loves it.
It was narrated from ‘Abdullah ibn Mas‘ud that the Prophet (blessings and peace of Allah be upon him) said: “There are three things that the Muslim should never fail to do: doing righteous deeds sincerely for the sake of Allah, showing sincerity towards the leaders of the Muslims, and staying close to the main body of the Muslims (jama‘ah), for their supplication will be answered.” (Narrated by Ahmad (12937) and at-Tirmidhi (2658); classed as sahih by al-Albani)
When the Messenger of Allah (blessings and peace of Allah be upon him) was asked, Which of the people is best, he said: “Everyone who is pure of heart and sincere of speech.” They said: Sincere of speech, we know what this is, but what is pure of heart? He said: “It is (the heart) that is pious and pure, with no sin, injustice, rancour or envy in it.” (Narrated by Ibn Majah (4216); classed as saheih by al-Albani in Sahih Ibn Majah)
This hadith encourages one to have a sound heart and to cleanse it of grudges and rancour.
How good is the believer’s condition, as he has a sound heart and is sincere to the people around him, and he calls them to Allah, bears their annoyance with patience and forgives them, and when he goes to bed he has no resentment or rancour in his heart towards anyone. It is a sufficient reward for patience that the patient person will be given his reward without measure on the Day of Resurrection.  al-Awza‘i said: [The reward] will not be weighed out or measured out to them; rather it will be scooped out for them in abundance. (Tafsir Ibn Kathir, 7/89)
So how about one who forgives and overlooks, then checks his heart and purifies it by means of faith, cleansing it of any impurity until it is completely pure?
We ask Allah, may He be exalted, to improve our attitude and to make us among those who possess this noble characteristic.
For more, please see these answers: 104919 , and 140798 . 
Also, see this category: The Heart Softeners 
And Allah knows best.