No, not at all! "It" would sound very strange and unfriendly. If it's a boy monkey, you use 'he', and if it's a girl monkey, you use 'she'. In this song, the monkeys happen to be male, so you use 'he'.

There's no rule. In the case of the monkeys, I would not use "it." The natural thing would be to assume that the monkeys are males and use the pronoun "he." A feminist, or someone trying to use gender-equal language... or someone teaching the poem to a little girl... might decide that some of the monkeys were girls and say "she." In the United States, we almost always use "he" and "she" for mammals whose sex is obvious--particularly when we are thinking of animals as having thoughts, emotions, feelings, or personalities. We use "it" when we actually don't know the sex (baby chicks, dolphins, insects, worms). We use "it" when we want to be emotionless about the animal (a psychologist describing a lab rat, for example). There are some interesting exceptions. In the United States it is very common for people to refer to dogs as "he" and cats as "she," because that seems to fit dogs' and cats' personalities--and because people don't want to stare at an animal's hind end to check!