Is it permissible to speak with a man who claims that the origin of man is a monkey, even though he knows about Islam, and would I bear a burden of sin if I continue to converse with him after he has said that?

The theory of human evolution is pure nonsense and falsehood, clashing with explicit verses from the Book of Allah, and it is a statement of disbelief (kufr). You bear no sin for what this person uttered. It is advisable not to engage extensively in argumentation with him unless you possess sufficient knowledge, for the heart is weak and doubts are snatchers. There is no problem in speaking with him for the purpose of giving sincere advice. As for shunning him, it is tied to the benefit; if you fear being influenced by associating with him, then distance is obligatory. Similarly, if shunning him leads to his rectification and return to the truth. Shunning varies depending on the strength and weakness of those shunning, and its goal is to deter and discipline the shunned person and for the general public to turn away from his state. If the benefit outweighs, such that it leads to a weakening of evil, then it is permissible; otherwise, it is not.