I don't think it's entirely necessary, though perhaps preferable for more serious songs. However, there are some perfectly acceptable songs where there are no rhymes: a friend wrote a song like this then, having sung it to us, asked people what was odd about the song - no-one had actually spotted that there were no rhymes!
Imperfect rhymes can add a bit of fun to comic songs, especially if the audience almost anticipates what the final word is going to be , and knows it will be a really "bad rhyme".