There is little to nothing a materialist would agree with from the Phaedo. This is because Plato's main argument in the dialog is that the soul is immortal, and this argument presupposes that there is a soul. The main content in the dialog are his four arguments for the immortality of the soul; if we take a materialist view and reject that the soul is immaterial, those four arguments are no longer relevant to the conversation because they all presuppose a dualism. The arguments might be used by a dualist to argue for dualism against a materialist, but the materialist will find noting to agree with in the arguments themselves if she retains her view.
The Greek word that is translated as "soul" is "ψυχή" ("psyche") which is believed to be "the immortal part of a person." Important to the argument is that the soul lasts after the body dies and to this extent Plato's idea of a soul is at least some form of dualism, even if it isn't exactly what DesCartes argued for.
Plato introduced his tripartite theory of the soul in the Republic. From the SEP:
The Republic also puts forward a new theory of soul, which involves the claim that the embodied human soul has (at least) three parts or aspects, namely reason, spirit and appetite.
In the Phaedo, Plato's main argument is that the soul is immortal and undergoes the process of metempsychosis, or transmigration of the soul, after death. He makes four arguments that he believes are so sound that any philosopher would see that they are true and would therefore not fear death in the same way Socrates did not. The four arguments are (paraphrased from the Internet Encyclopedia of Philosophy):
The cyclical argument: the Forms of everything, including souls, are eternal and unchanging and everything comes to be from it's opposite state. Opposite states are created by opposite processes, life and death are opposite states and are created by coming to life and dying. If there wasn't balance to the opposite states then eventually everything would be in one state, but that contradicts that all Forms are eternal, therefore souls have to be immortal and come back to life.
The argument from recollection: Humans possess some sort of a priori knowledge that doesn't come from sense experience. This can be demonstrated by asking people with severely limited knowledge of geometry questions in certain ways as to allow them to figure out the answer a priori. The only way that we could have prior knowledge is if our souls are eternal, have knowledge of Forms and we merely recollect this knowledge in life.
The affinity argument: There are only two things that exist, material things like those found in the world and immaterial things such as Forms. The body is a worldly material thing which is like physical objects, but our soul is immaterial and divine like Forms. Therefore, because Forms are eternal, the soul must be eternal too as it is not like material, physical things.
The final argument: Forms are eternal and every single thing participates in Forms. This means that every thing that can exist, even immaterial things such as the soul, possess the quality of some Form or another. The soul participates in the Form of life, because it is here when we are alive. Forms are unchanging and things that participate in Forms do not stop participating in those Forms, therefore the soul will never stop participating in the Form of life, which is to say it is immortal.
Plato uses all four of these arguments for the immortality of the soul. Central to all of these arguments is that the soul is something immaterial and separate from our body. These four arguments are the central arguments given for the entire dialog and a rejection of the soul being immaterial, or even existing, would leave the arguments with nothing to say.
To give an illustrative example, let's try to substitute in something that a materialist would argue for into Plato's arguments and see what is left. For this, let's imagine this philosopher believes the mind is just the consciousness brought about by neural activity in the brain.
The cyclical argument would become an argument that someone's neural activity would need to eventually start to happen again after their death because opposites need to balance each other out. A materialist would be more likely to have their views of causation in line with what modern science says and this argument doesn't seem to hold any scientific weight. Irreversible damage to the neural structure of a body as it decays would not reverse itself just because "nature needs to be balanced."
The argument from recollection might be seen as a "God of the gaps" argument from a materialist. Plato says that there is no way that we could have a priori knowledge without our souls retaining it from another life. However, current understandings of things like language and psychology allow for psychological nativism which can explain a priori justification and knowledge.
The affinity argument explicitly states that the souls is an immaterial object with properties similar to other immaterial things. If we accept the materialist view that the brain is the source of consciousness then we would also accept that the mind dies when the body dies, as the neurons stop functioning. This is a direct contradiction of what the argument asserts.
Finally, the the final argument could have a similar rejection to the rejection of the cyclical argument. The argument would be that the neurons in someones brain have been alive and functioning at one point and according to the theory of Forms they would have to continuously function because Forms are unchanging. This is in direct contrast to what we know about the body. Plato himself would even reject this idea. He sees the brain as being part of the body and the brain dies along with the body. If we say that the brain is what creates the mind then the mind would need to die as well. He wouldn't be happy with saying the brain is what creates the mind but he would argue if that were true then the mind would die along with the body.