Is it permissible for me to work for an illiterate man who wants me to manage his electronic investments – knowing their impermissibility – in exchange for a fixed wage, and regardless of whether the profit is lawful or unlawful?

If a man asks you to work in electronic trading that is not Sharia (definition)-compliant, or in any forbidden work, then this is impermissible, and your wage would be unlawful. However, if the work is permissible, such as tailoring, but all of his money is unlawful, then it is not permissible to work with him. But if his money is mixed between lawful and unlawful, then working with him is permissible, though disliked.