C++ error: invalid initialization of non-const reference of type char&

For starters, if you're already writing in C++, use idiomatic containers. No char[][][][], of which some parentheses indicate that it is a string, and others indicate that it is an array.

Then, the three elements "Alex", "22-33-44", "24300" are not just three strings of some kind, they are things with different semantics, and are obviously combined into one structure.

Then, in your code, you define here and there some magic constants: 30, 120, etc. I hope you understand, what it threatens. If not, think for yourself.

Now for a specific error. According to your code, the displayWorkers procedure requires the first argument to be char& - that is, a reference (not a pointer) to char (that is, a single character). You both pass and use the workers structure inside.

Your code should look something like this:

#include <iostream>
#include <vector>

using namespace std;

/* types */
struct worker {
    string name;
    string phone;
    unsigned salary;
};

/* func declarations */
void displayMenu();
void displayWorkers(const vector<worker>&);

/* global variables */
vector<worker> workers = {
    { "Alex", "22-33-44", 24300 },
    { "Ivan", "55-33-66", 30000 },
    { "Pavel", "22-33-44", 24300 },
    { "Антон", "", 25000 },
    { "Жора", "44-33-11", 25000 }
};

/* functions */

int main() {
    string listen;
    // show menu
    do {
        displayMenu();
        getline(cin, listen);
        if (listen == "1") 
            displayWorkers(workers);
        } else if (listen == "2") {
            cout << " -- Projects --\n";
        }
    } while (listen != "quit");
    return 0;
}

/**
 * Display menu
 */
void displayMenu() {
    cout << " == Menu ==\n";
    cout << " 1 - Показать всех работников\n";
    cout << " 2 - Показать все проекты\n";
    cout << " quit - Выход\n";
}

/**
 * displayWorkers
 * @param workers
 */
void displayWorkers(const vector<worker>&workers) {
    cout << "Workers\n Name\t\t|Phone\t\t|Salary\n";
    for (auto& w : workers) {
        cout << w.name << "\t\t|" << w.phone << "\t\t|" << w.salary << "\n";
    }
}

Verification: http://ideone.com/YOdMsy

This is

char workers[][2][30] = {
    "Alex", "22-33-44", "24300",
    "Ivan", "55-33-66", "30000",
    "Pavel", "22-33-44", "24300",
    "Антон", "", "25000",
    "Жора", "44-33-11", "25000"
};

Defines a 3-dimensional array of type char, which you are trying to fill in the style of a one-dimensional array of type const char*. Judging by the data you use, you need a structure and a one-dimensional array. If it's C++, then you can do it, for example, like this: The structure itself:

struct Employee
{
    std::string m_Name;
    std::string m_Number;
    std::string m_AnotherNumber;

    Employee(const std::string& name, const std::string& number, 
        const std::string& anotherNumber) : 
        m_Name(name),
        m_Number(number),
        m_AnotherNumber(anotherNumber)
    {

    }
};

Using:

Employee workers[] = {
    Employee("Alex", "22-33-44", "24300"),
    Employee("Ivan", "55-33-66", "30000"),
    Employee("Pavel", "22-33-44", "24300"),
    Employee("Антон", "", "25000"),
    Employee("Жора", "44-33-11", "25000")
};

std::cout << workers[2].m_Name << "\n";

DisplayWorkers, respectively, will look like this:

void displayWorkers(Employee workers[], int count) 
{
    std::cout << "Workers\n Name\t\t|Phone\t\t|Salary\n";
    for(int i = 0; i < count; i++) {
        std::cout << workers[i].m_Name << "\t\t|" << workers[i].m_Number << "\t\t|" 
            << workers[i].m_AnotherNumber << "\n";
    }
}

Well, since C++, then let's add the example to normal C++ decisions:

#include <iostream>
#include <string>
#include <vector>
#include <iomanip>

class Employee
{
public:
    Employee(const std::string& name, const std::string& number, 
        const std::string& anotherNumber) : 
        m_Name(name),
        m_Number(number),
        m_AnotherNumber(anotherNumber)
    {

    }
    std::string name() const
    {
        return m_Name;
    }

    std::string number() const
    {
        return m_Number;
    }

    std::string anotherNumber() const
    {
        return m_AnotherNumber;
    }
private:
    std::string m_Name;
    std::string m_Number;
    std::string m_AnotherNumber;
};

void displayWorkers(const std::vector<Employee>& workers) 
{
    size_t fieldSize = 15;
    std::cout << std::setw(20) << "Workers\n";
    std::cout << std::left << std::setw(fieldSize) << "Name" <<
        std::setw(fieldSize) << "Phone" << std::setw(fieldSize) << "Salary";
    std::cout << "\n\n";
    for(auto worker: workers) {
        std::cout << std::setw(fieldSize) << worker.name() << std::setw(fieldSize) 
            << worker.number() << std::setw(fieldSize)
            << worker.anotherNumber() << "\n";
    }
}

int main()
{
    std::vector<Employee>  workers = {
        {"Alex", "22-33-44", "24300"},
        {"Ivan", "55-33-66", "30000"},
        {"Pavel", "22-33-44", "24300"},
        {"Антон", "", "25000"},
        {"Жора", "44-33-11", "25000"}
    };

    displayWorkers(workers);

    return 0;
}

It will be given to you as a homework assignment.