Question: Do I need to use **any** or **the** before **other** in the following sentence when comparing things?

Here are the examples: 

* Moscow is bigger than **any other** city in the world.
* Moscow is bigger than **other** cities in the world.
* Moscow is bigger than **the other** cities in the world.

Some more examples:

* John is taller than **other** boys.
* John is taller than **any other** boy.

Answer: 
All of your sentences are grammatical, whether **other** is preceded by **any**, **the**, or nothing at all.

I, personally, find the second sentence the most natural, followed closely by the first, and, finally, the third. But all sound quite natural to me nonetheless.

Of course, that's subjective. Different people will have different reactions to them.
Yes, you do. Without 'any', 'the', 'other' it would be quite vague as to what the comparison is referring to. 

Note: bigger is a bit vague. Do you mean big is in land mass, population? etc.