This sounds like a very interesting and possibly complex undertaking. I am assuming that you would be self-publishing this, so my answer will be based on that assumption.
The first step you will need to take is to identify who will be the editor, which is the title that would be assigned to the person responsible for all the work involved in making this project happen. While it may be beneficial to identify a few people who can help go through the initial selection process, I would recommend having one final editor who is responsible for the final selection, editing, and formatting of the poems that will be included.
How you go about obtaining the poems you need for your anthology could take several different approaches. The easiest would be to approach this as an opportunity for each writer to have their work published as part of an anthology. If you state that you are willing to include their poems in exchange for the one-time publishing rights, then I'm sure you will find that there will be a good number of poets willing to accept those terms. This would grant the editor the legal right to publish the poems while still permitting the poets to retain their copyright privileges for their work. If you state these terms as conditions for inclusion, then that should be sufficient enough for protecting yourself from a legal perspective.
Once you have selected the poems you wish to include in your anthology, the next step would be the actual publishing process. You could use a service such as Kindle Direct Publishing to easily self-publish the anthology in e-book format. You could also use a service such as CreateSpace to make the book available in print format. There are a number of other services you could use, but I personally prefer these two.
This step is where it becomes more important to have just one person who is ultimately responsible for handling things. Each of these self-publishing services requires that you create an account for managing any work that you publish through them. Most of them will only allow one person to manage that account, and when it comes to paying royalties, they will usually pay them to only one person. If you want to do this as a group, then one person will have to take responsibility for distributing the funds, and that could become quite a major undertaking.
Since the editor is the person doing most of the work, most people would not begrudge him/her receiving payment for that effort. However, people I have known who have undertaken such an effort have decided to donate the proceeds from their anthologies to a charity that is identified in advance. Some might retain enough money to offset any costs they may have incurred, but that is ultimately up to you.
If you would prefer to buy the rights to each individual poem in advance in order to provide the poets with some form of compensation, then you could use a service like Kickstarter to raise the funds for that. You could also use it to pay for having the book professionally edited and for the design of a book cover, but that would be a slightly larger undertaking. Ultimately it will depend on how much effort you really want to put into this project.