Most likely issue comes form Substitue operation which requires as per instructions: 

"If the raw password contains the given substring, replaces all of its occurrences with the substitute text given and prints the result."

Without a global regex set in the replace function, replace will only modify the given string at the first occurence of the given substring.

MDN: String.replace() => Note that the function will be invoked multiple times for each full match to be replaced if the regular expression in the first parameter is global.

Test with following input/output to see where problem comes from:

password((['up8rgoyg3r1atmlmpiunagt!-irs7!1fgulnnnqy',
    'TakeOdd',
    'Cut 18 2',
    'Substitute ! ***',
    'Substitute ? .!.',
    'Done',])
);

Output
programming!is!funny
programming!is!fun
programming***is***fun
Nothing to replace!
Your password is: programming***is***fun

Demo:

function solve(input) {
    let password = input.shift();

    let line = input.shift();

    while (line !== 'Done') {
        let [command, ...rest] = line.split(' ');

        switch (command) {
            case 'TakeOdd':
                let newPassword = '';

                for (let i = 0; i < password.length; i++) {
                    if (i % 2 !== 0) {
                        newPassword += password[i];
                    }
                }
                password = newPassword;
                console.log(password);
                break;
            case 'Cut':
                let index = Number(rest[0]);
                let length = Number(rest[1]);

                let string = password.substring(index, index + length);
                password = password.replace(string, '');
                console.log(password);
                break;
            case 'Substitute':
                let stringTarget = rest[0];
                let replace = rest[1];
                if (password.includes(stringTarget)) {
                    while (password.includes(stringTarget)) {
                        password = password.replace(stringTarget, replace);
                    }
                    console.log(password);
                } else {
                    console.log('Nothing to replace!');
                }

                break;
        }

        line = input.shift();
    }

    console.log(`Your password is: ${password}`);
}

Здравей, LilianaVelvheva,

Проверката за това дали паролата съдържа подадената от отвън част не може да се направи така, както ти си я направила. Трябва да бъде

if (!passwords.includes(elementToReplace))

Колегата Axiomatik вече е дал решение на проблема, а аз мога да предложа алтернатива за процеса на заместването със следния израз, който също работи:

passwords = passwords.split(elementToReplace).join(elSubstitute);