#Preface
First, this question assumes a level of homogeneity that Rastafarianism -- like many religions -- does not have. It's a bit like asking about the impact a given Pope had on Christianity; you'll get a very different answer answering from the Catholic point of view than you would from the Protestant point of view.
Still, there is validity to the question, because despite the varied doctrinal differences encapsulated within Rastafarianism, one of the universal tenets is the divinity of Haile Selassie.
#Selassie and Rastafarianism
Before discussing the impact of Selassie's overthrow and death on the religion, let's discuss what Selassie meant to Rastafarianism prior to that. Not that I wrote "divinity" above, rather than "god incarnate" as your question asks, because the specifics of that divinity were in flux over the evolution of Rastafarianism. He was God incarnate, he was another in the line of Jesus, he was the Holy Spirit.
Certainly, there's no denying the central importance of Haile Selassie to Rastafarianism. Indeed, he is the source of the religion's name: prior to becoming emperor, he was Ras Tafari (Ras an Amharic word for a Nobel or Prince).
It was Selassie's crowning as the 225^th Emperor of the line of King Solomon, "King of Kings and Lord of Lords, Conquering Lion of the Tribe of Judah, Elect of God, and Light of the World" that sparked the beginning of Rastafarianism.
The religion did not appear ex nihilo, however. Prior to Selassie, the figures of Shaka Zulu (creator of the Zulu Empire) and Jaja of Opobo (a Nigerian rebel against British colonialism) had long inspired Jamaican slaves.
Also important was Marcus Mosiah Garvey, whose "Back to Africa" movement was a seed for the later religion. He is said to have told his followers in 1920 to "Look to Africa, when a black king shall be crowned, for the day of deliverance is near." Selassie was recognized as that King by a few Christian ministers: "Leonard Howell [...], Archibald Dunkley, Joseph Hibbert, and Robert Hinds."
One of the religion's founding tenets, as expressed by Howell, was the "the acknowledgement of Selassie as Supreme God and the Ruler of all Black people." This did not entirely break from the Christian tradition: Selassie was the most recent and most great of the "manifestations of a greater divinity, regarded as Jah in the flesh" -- but Jesus Christ and the Prophet Elijah were previous examples of such an incarnation.
#Selassie's Overthrow and Death
On September 12, 1974, after 58 years of rule, Emperor Haile Selassie was placed under arrest. The circumstances leading up to this coup d'etat are worth a post of their own (and would fit perfectly with this week's theme!)
The coup was a not sudden, but a slow process. Tracing it through the New York Times we see the headlines slowly escalate: on July 3, "Ethiopia's Armed Forces Isolating Haile Selassie"; on July 10, "Ethiopian Cabinet Gets Army Guidelines"; on August 7, "A New Constitution Offered in Ethiopia"; on August 17, "Ethiopian Military Strips Selassie Of Some Powers"; on August 27, "Selassie's Wealth Reportedly Seized"; and finally on September 13, "Haile Selassie is Deposed by the Ethiopian Military after Ruling for 58 Years."
The military reported Selassie dead on August 27, 1975. They did not, however, produce his body.
#Rastafarianism and Selassie's Death
The removal of Selassie had a number of short-term effects on Rastafarianism. The Rastafarian Jamaicans who had moved to Ethiopia to settle in Shashemene, in an area that Selassie had granted to "the black people of the world," certainly experienced some repression. Giulia Bonacci interviewed the Rastafarian Junior Dan in 2003, and I'll quote that interview for a first-hand account of the experience:
BONACCI: During the Derg was there any pressure about His Majesty?
JUNIOR DAN: Yes, there was pressure. We couldn't talk anything about His Majesty, we couldn't display portraits of His Majesty, nothing where Haile Selassie and his government was concerned. You have to be totally conservative. Through those years, although we were publicly and openly declaring our faith and what we stand for, we weren't pressured in ways like individually by the government, we didn't get any molesting from them. They didn't disturbed us. I can say truthfully speaking the Derg government, where the Rasta community is concerned, is much better than this present government.
Selassie's death was not immediately accepted by Rastafarians. Bob Marley's "Jah Live," released a few days after Selassie's reported death, summarized the views of many:
Fools saying in their heart
"Rasta, your God is dead"
But I and I know, Jah Jah!
Dread, it shall be dread a dread
Jah live! Children yeah!
Jah Jah live! Children yeah
This view persisted at least through 1988, when Van Dijk wrote about the views of The Twelve Tribes of Israel, one Rastafarian sect:
The Twelve Tribes absolutely deny the death of the Emperor in August
(All emphasis my own).
#Rastafarianism After Selassie's Death
As it did sink in to the Rastafarian consciousness, it represented a theological challenge -- but not one that couldn't be overcome. One interpretation was that the death marked "Haile Selassie [as] the true Messiah, 'in the flesh until August, 1975, but in the spiritual body since his death.'"
Moving into the 21st century, Rastafarianism has 3 main sects: the Twelve Tribes of Israel, the Bobo Shanti, and the Nyabinghi. In 2000, Selassie, whose "bones [had been] found under a concrete slab on the grounds of his former palace in 1992," was buried. This was an "emotional ceremony, in which members of the [Ethiopian Orthodox Church], Rastafarians (including Rita Marley, the widow of Bob), and many others gathered to honour his memory."
Chakravarty offers perhaps the most formalized framework for understanding Rastafarianism, based on research and interviews from 2008 through 2011. He identifies four types of Rastafarianism: Orthodox-Isolated Rastafari, Orthodox-Integrated Rastafari, Secular-Spiritual Rastafari, and Secular-Cultural Rastafari. Each group continues to greatly revere Selassie, though secular-cultural -- labelled by some as "pseudo-Rastafarian" -- includes those who "who deny the divinity of Haile Selassie and regard him as just another great human being." Note that these types should not be perceived to constrain or essentialize Rastafarianism, rather, to help organize our thoughts on the religion's immense diversity. Certainly there's more to be said on Rastafarianism in the 21st century, but that grows removed from Selassie and probably belongs in another question.
#Citations
Charet RM. Jesus was a Dreadlocks: Rastafarian Images of Divinity. Sydney Studies in Religion. 2008 Nov 10.
Harding W. Dubbin'the Literary Canon: Writin'and Soundin'A Transnational Caribbean Experience (Doctoral dissertation, Oberlin College).
Bonacci G. From Pan-Africanism to Rastafari. Hurst; 2015.
Bonacci G. The return to Ethiopia of the Twelve Tribes of Israel. New West Indian Guide/Nieuwe West-Indische Gids. 2016 Jan 1;90(1-2):1-27.
Bonacci G. An interview in Zion: the life-history of a Jamaican Rastafarian in Shashemene, Ethiopia. Callaloo. 2011;34(3):744-58.
Cusack CM. The Romance of Hereditary Monarchs and Theocratic States: Ethiopia and Emperor Haile Selassie I in Rastafarianism and Tibet and the Fourteenth Dalai Lama, Tenzin Gyatso, in Western Buddhism. Alternative Spirituality and Religion Review. 2013 Apr 1;4(1):122-46.
Koehn P. Ethiopian Politics: Military Intervention and Prospects for Further Change. Africa Today. 1975 Apr 1;22(2):7-21.
Dijk FJ. The twelve tribes of Israel: Rasta and the middle class. New West Indian Guide/Nieuwe West-Indische Gids. 1988 Jan 1;62(1-2):1-26.
The Associated Press. An Imperial Burial for Haile Selassie, 25 Years After Death. 2000 Nov 6.
Chakravarty KG. Rastafari Revisited: A Four-Point Orthodox/Secular Typology. Journal of the American Academy of Religion. 2014 Dec 23;83(1):151-80.
The New York Times. Various. Accessed via Times Machine.
P.S. Thanks to the Mods, in particular /u/Elm11, for their excellent work in keeping this thread, and the subreddit, clean.