Oh, jeez. Well, without having a specific person in mind that you want to run a DNA test on, there’s a lot of inbetween-ed-ness in here that most people tend to miss.

The tl;dr is that it is possible. There has never been a wholesale replacement of the population that would cause a genetic break with the ancient Egyptians (I have seen a lot of uninformed posts to the effect that there is no relationship because ‘the Muslims killed all the ancient Egyptians and occupied their country,’ and that’s just completely incorrect). Most Egyptians are probably the descendants of multiple groups, given the country’s location at a major crossroads between Europe, Asia, and Africa.

Remember, Ancient Egypt lasted for over five thousand years. At various points, Hittites, Babylonians, Nubians, Persians, and Greeks all came and conquered; we’re also talking about the common people, so we need to take into account traders from distant lands who came to do business and intermarried.

Egypt was incorporated into the Roman Empire after the death of Cleopatra VII (who didn’t look anything like Elizabeth Taylor); Rome itself was a multi-ethnic empire, which brought in a whole new set of DNA to the mix.

The Arab armies arrived in the winter of 640-641; they found relatively willing accomplices among the local Coptic Christian population who had been excommunicated wholesale at the Council of Chalcedon and the Byzantine governors occasionally conducted massacres and purges of them. Records are a bit scarce from the time, but the Arabs appear to have been mostly urban dwellers; there was not much settlement in agricultural areas, where the bulk of the Egyptian population lived (and continues to live).

The Coptic language, which is the surviving linguistic descendant of the ancient Egyptian language, was still spoken in rural villages as late as the 18th century (it’s still the liturgical language of the church).

Then, of course, you have 15 centuries of people coming back and forth during the Islamic period. Cairo was one of the largest cities in the world around the first millennium, and a noted center of learning, and a common stopping point for people coming from the west en route to Mecca for the pilgrimage.

There are also now Nubians settled throughout the country, both from labor migration as well as displacement by the various dams built on the Nile in the 20th century.

All of which is a long winded way of saying there is no particular reason that a modern Egyptian could not be a descendant of the ancient Egyptians; there has been a significant amount of migration in and out, so many of them will have genetic roots in Egypt and Arabia, and probably a number of other places as well.

Huh, I will be presenting at a conference this weekend on nearly exactly this topic. Here's an Egyptian indigenous theory perspective on this issue.

It matters first of all how you intend to measure ethnicity. Ethnicity does not inherently have any biological component -- it is largely a matter of identification and group membership. Some oberservers would attempt to answer this question assuming you mean to find a biologically traceable, and thus ostensibly objective, basis for measuring both ethicity and specifically the ethnicities of living Egyptians. But genetics can do little, if anything, with regards to locating identification and group membership. That is something you have to do ethnography and documentary analysis to find. If what you are looking for is indeed how living Egyptians identify relative to Arabian and older Egyptian ancestors -- and not, as a great many non-Egyptian observers have sought since European invasion, a reason to justify past and continue ongoing, material disenfranchisement of living Egyptians from their own ancestors and histories -- then you cannot look to genetics for your answer.

The direct precedent to my work is Lois Keck's 1984 ethnography of Egyptians living in the DC area (it's been kind of a while since anybody did some serious ethnography of Egyptians in the area). In that project, she found that while many of those Egyptians did express Arab, "Middle Eastern," Muslim, and Christian identifications, they always reminded her that the core of their Egyptian identities was what she described as "a psychological identity with the land," and a primary relationship with Pharaonic ancestors and roots.

Unfortunately it seems many living Egyptians take their indigeneity for granted and have either a hard time articulating it to non-Egyptians or simply feel little urgency to doing so. Al-Aswad, for example, has described Egyptians as indigenous almost in passing, and Ahmad Kamal wrote of himself as "the sole indigenous Egyptologist" in an appeal to avoid the placement of another European at the head of the antiquities service near the turn of the 20th century. A racialized, genetics-focused interpretation of possible Egyptian indigeneity would attempt to disqualify Kamal's claim due to his Turkish ancestry, but a more nuanced indigenous theory reading would be able to process Kamal's claim as an example of the vast diversity and complexity that indigenous people experience and live in their communities every day. I suspect the mechanics of this reading are beyond your question.

Most mundanely, exceedingly few living, self-identifying Egyptians also identify with any Arabian tribal groups, which are a critical measure of Arabian ethnicity.

Bottom line: from at least one indigenous theory perspective, Egyptians are Egyptians. Being Egyptian necessarily involves incredibly complicated and old relationships with a variety of histories, and ethnographic evidence largely indicates that their (and now I should say, our) most ancient ancestors are among those histories, and perhaps even the highest priority among them.

To better answer the question, one may ask, did the fall of Egypt to Arab rule cause a demographic change? The answer is yes.

the Arabs expansion into Egypt, in the seventh century, resulted in a significant number of Arabs moving there. Not only did Arab control over Egypt cause a demographic change, it also reoriented the political and economic center of Egypt from Alexandria to the new city the Arabs founded, Fustat (see Jelle Bruning's The Rise of a Capital: Al-Fusṭāṭ and Its Hinterland). This new city would grow significantly in size and importance, and next to which a new capital would be built in the tenth century, Cairo. So, the seventh century was the staring point of Arab presence in Egypt that continued to grow from then onwards.

However, as u/khowaga has noted, the inhabitants of Egypt before the Arabs took over were not displaced, rather they continued to live adapting to new rulership. They, along with the Arabs, would adapt to new rulers, the Turks, the French (briefly) and the British.

A good book that may be of interest on Egypt (though it mostly focus on the history of the site of modern Cairo) is Cairo: Histories of a City by Nezar AlSayyad.