First, historically speaking, many poets (who are very serious and clearly poets) have written in blank verse - poetry which has a particular meter (iambic pentameter) but no rhyme scheme. We associate this usually with drama - in particular Shakespeare, but also Christopher Marlowe. However, the form has also been widely used in poetry as well - John Milton's Paradise Lost is written in blank verse, and the Romantic poets (notably Byron, Shelley and Keats) all wrote in blank verse. Hopefully as the blog progresses I will look more closely at various poetic forms, so I won't say too much more about blank verse at this stage.
Of course, blank verse does have a particular rhythm (not that any poet completely respects the rhythm - more on this in the as yet only potential post on rhythm). So the question could be revised to 'Does poetry have to have rhyme and/or rhythm'? If by rhythm you mean an actual meter, like iambic pentameter, then the answer is no.
Here is a poem by William Carlos Williams, 'This is just to say', which doesn't rhyme at all.
I have eaten the plums that were in the icebox and which you were probably saving for breakfast Forgive me they were delicious so sweet and so cold
This poem could be written out without the line breaks, as a note from a husband who came home late to his wife. But would such a husband have written it like this? More like he would have written: 'Sorry, I ate all the plums in the icebox, I hope you weren't saving them for breakfast.'
WCW's poem draws out each separate part of the moment - 'the plums' and 'the icebox' each on a separate line in the first stanza emphasises the two central objects. 'I have eaten' is an almost imperious intoduction to a simple poem about a simple moment. In the second stanza, the introduction of 'you' conjures a relationship - someone with whom the writer shares the icebox, a life. The intimacy conjured by his knowedge of 'you', saving (once again the separation of the word conjures 'saving' as a special, thoughtful act). Then forgiveness is asked, not because the writer was hungry or tired or had a big day, but because of the particular deliciousness of the plums - each element 'so sweet/and so cold' separated out. I could talk about this poem for much longer, because I love it, but I will leave my example there.
And this is why poems don't have to rhyme - the use of line breaks and/or the natual rhythms and order of language can conjure so much (more on line breaks when we get to prose poetry). A poem can slightly alter natural spoken rhythms for emphasis, place special importance upon ordinary things by highlighting particular words. So 'This is just to say' is a note apologising for eating the plums, but it is also about a simple and beautiful moment.
And I haven't even made it to Walt Whitman yet.