The required amount of Remembrance of Allah (dhikr) is as much as possible
Allah ordered that He should be remembered abundantly. Describing the wise men and women who reflect on His signs, the Quran mentions:
Those who remember Allah standing, sitting and on their sides (3:191).
Those men and women who engage much in Allah's praise. For them has Allah prepared forgiveness and a great reward (3:191, 33:35).
The author of Fiqh al-sunna says that Mujahid explained: "A person cannot be one of those men and women who remember Allah much as mentioned in the above verse of the Quran, unless he or she remembers Allah at all times, standing, sitting, or lying in bed." He also says that when asked how much dhikr one should do to be considered one of "those who remember Allah much" (33:35), Ibn as-Salah said that "much" is "when one is constant in supplicating, in the morning and evening and in other parts of the day and the night as reported from the Prophet (saw).”
Concerning the above Quranic verses, Ali ibn Abi Talha relates that Ibn Abbas said:
All obligations imposed upon humanity by Allah are clearly marked and one is exempted from them only in the presence of a genuine cause. The only exception is the obligation of dhikr. Allah has set no specific limits for it, and under no circumstances is one allowed to be negligent of it. We are commanded to "remember Allah standing, sitting and reclining on your sides" (3:191), in the morning, during the day, at sea or on land, on journeys or at home, in poverty and in prosperity, in sickness or in health, openly and secretly, and, in fact, at all times throughout one's life and in all circumstances.
It is clear through the above evidence that there is no such thing as too much dhikr. The Prophet (saw) said, "He who loves something mentions it much."46 Those who love Allah and His Prophet (saw) mention Allah and His Prophet (saw). No one will limit this practice except those who do not feel such love. Imam Ghazali said:47
It is person's soul and spirit that constitute his real nature ... Upon death his state changes in two ways. Firstly he is now deprived of his eyes, ears and tongue, his hand, his feet and all his parts, just as he is deprived of family, children, relatives, and all the people he used to know, and of his horses and other riding-beasts, his servant-boys, his houses and property, and all that he used to own. There is no distinction to be drawn between his being taken from these things and these things being taken from him, for it is the separation itself that causes pain ...
If there was anything in the world in which he had found consolation and peace, then he will greatly lament for it after he dies, and feel the greatest sorrow over losing it. His heart will turn to thoughts of everything he owned, of his power and estates, even to a shirt he used to wear, for instance, and in which he took pleasure.
However, had he taken pleasure only in the remembrance of Allah, and consoled himself with Him alone, then great bliss and perfect happiness will be his. For the barriers that lay between him and his Beloved will now be removed, and he will be free of the obstacles and cares of the world, all of which had distracted him from the remembrance of Allah. This is one of the aspects of the difference between the states of life and death.
On the same topic Imam Habib al-Haddad said:48
Time and days are a person's capital, while his inclinations, desires, and various ambitions are the highway robbers. The way in which one profits on this journey lies in succeeding in coming to Allah and in attaining everlasting happiness, while one loses by being veiled from Allah, and being consigned to the painful torment of the fire.
For this reason the intelligent believer transforms all his breaths into acts of obedience, and interrupts them only with the remembrance of Allah (dhikr).
(The next page has additional hadith relating to dhikr.)
46 Narrated by Abu Nuaym in the Hilya and Daylami in Musnad al-firdaws. Sakhawi cites it in al-Maqasid al-hasana p. 393 #1050 and does not comment upon it.
47 Imam Ghazali, in the fortieth book of his Ihya entitled "The Remembrance of Death and The Afterlife" (p. 124 in the translation of T.J. Winter).
48 Imam Habib al-Haddad, Key to the Garden p. 104.
(May Allah (swt) forgive me for any errors I may have introduced through transcription or editing of this text.)