I apologize that I mostly know about Coptic and Berber but I will try to demonstrate why this question is hard to answer and then introduce a recent theory into this particular subject. First, I would like to point out that when talking about "Berber" or "the Berber" one is actually talking about quite a diverse group of people and languages, some of which are not around today and some which still exist. I will reference "the Berber" in as general terms as I can but understand there are exceptions and differences among them. Additionally, there is not as much research on the Arabs and the Berbers in context with linguistic shift as there is for Coptic and Persian.
One is tempted to believe that conversion to Islam would inherently coincide with adoption of Arabic but those two factors actually don't share as close of a link as you would expect. Coptic Christians were using Arabic in their own Bibles as early as the 10th century^(2) They continued to be Christian and speak Arabic for the next century and to a lesser extent thereafter. Coptic Christian scholars eventually complained that Christians couldn't speak their own language in the 11th century^(3) Many Berbers converted to Islam quickly and revolted against the Arabs as early as 749 under their own form of Khawarij Sufrism yet their languages proved more resilient than Coptic over time (likely in great part due to the formation of the Almoravid dynasty)^(4) . Many early Muslim scholars would be angry with me for calling many Berbers of the Middle Ages "Muslim" since they often practiced syncretic forms of the religion but it's still a strong point in demonstrating separation between the religion and the Arabic language. Meanwhile, Persia (correction, see comment: Main Persian speaking area) became Islamisized quicker than either of those locations (early 8th century, there is some controversy to this, though) and yet the Persians managed to hold onto their language^(5). These cases demonstrate that conversion to Islam is a tricky factor in determining the cause of local languages extinction in comparison with one another.
One is also tempted to believe that the level of Arab migration an area received would influence the vitality of local languages. Indeed, the decline of the Coptic language in the 10th and 11th centuries coincided with increased Arab immigration yet it also coincided with a shrinking Coptic population, decreased heretic oppression, and an expanding economy under the Fatimids^(6 7). I will also concede that the Berber languages in North Africa do not seem to have declined so greatly until that area received higher migration but I also want to point out that Persia actually received more Arab migration in its earlier years than Egypt^(1)
Language replacement and language adoption are two different things. After all, bilingual societies exist around the globe and Coptic itself persisted under centuries of Greek rule when Greek became a dominant language in Egypt. Why do some societies exist bilingually and others not, though? Many factors come into play when considering language shift. Recently, scholars by the names of Reza Ghafar Samar and Tej K. Bhatia have released a study proposing that surface level structural similarity between two languages greatly influences the development of a bilingual society or a monolingual society where one language overcomes another. In a recent article released last year the two scholars attempted to explain why Arabic overcame Coptic while Persian persisted by comparing the structural similarities of the two languages^(5). They worked off the premise that when two languags come into contact they demonstrate "borrowing" (the wholesale importation of individual words or phrases from one language into another, often with target language influence) or "code switching" (switching between languages completely.) They argue that the more structurally similar two languages are the more likely speakers of one language who most often encounter both are to participate in code switching instead of just borrowing. They demonstrate that Arabic and Coptic share many similarities that Persian does not and therefore Coptic speakers would have participated in code switching moreoften than just borrowing. They argue that code switching contributed to language extinction and that this phenomenon can even be seen in surviving Coptic texts.
Therefore, Persian persisted into today in great part to its lack of structural similarity with Arabic. That also explains its abundant borrowing.
Many more details exist in their article and I encourage you to read it since they obviously explain it better than I can. I hold a small amount of skepticism towards their article, though, because I do not believe they properly delineated MSA from historical dialectal Arabic found in Persia vs. Egypt. Additionally they make no mention of Berber vs. Arabic.
Additionally I would like to mention looking into the Shu'ubiyya movement and the pro-Persian practices of the Abbasids but I will let those better versed in Persian history expand on those .
Predictability of language death: Structural compatibility and language contact RezaGhafar SamarabTej K.Bhatiac
Sullivan, Shaun. "Coptic Conversion and the Islamization of Egypt." Mamluk Studies Review 10, no. 2 (2006): 6579. http://mamluk.uchicago.edu/MSR_X-2_2006OSullivan.pdf.
Parker, Kenneth S. "Coptic Language and Identity in Ayyūbid Egypt1." Al-Masāq 25, no. 2 (2013): 222 39. doi:10.1080/09503110.2013.799953.