Here's my simplified version of Toby's original solution. This solution takes advantage of Jasmine's built-in toThrow() matcher which is similar to matchers in the should.js, expect.js, and chai assertion libraries which all work with Mocha.

omit-callback-unit-test.js

/*
The implementation below assumes you're using Jasmine's built-in matchers. But similar matchers
exist in the should.js, expect.js, and chai assertion libraries which work with Mocha.
*/

var makeSomething = function(callback) {
    // Below is the simplest implementation that will result in the test passing.
    // If you want a more robust implementation, you'll want to write more tests first. :)
    throw(new Error("Missing parameters"));
};

describe("makeSomething()", function() {

    describe("when called with no callback parameter", function() {

        it("should throw a meaningful error", function() {

            expect(function() {
                makeSomething();
            }).toThrow(new Error("Missing parameters"));

            // Note, you can also use toThrow() with no params if you don't care about matching
            // the exact error message.

        });

    });

});