How to flatten posters
What you will need:

Clean, flat surface
Clean, flat stiff sheet for top
Sheets of thick absorbent "buffer sheet" printmaking paper, about same size as posters. The paper should be acid-free, such as this Strathmore type.
They can be reused indefinitely. You can use multiple "sandwiches" over a large area.
Water spray bottle
Weights to flatten pile

Dry paper will not change shape; the trick is to gently introduce moisture into a set of posters as you flatten them.
What you don't want to do is get water on the poster itself, so this technique requires creating little "water sandwiches" to gently rehydrate the posters without damage.

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Created 4/12/2020, updated 4/23/2021

rolled posters
lay under sheet
wet buffer sheets
weigh down
Carefully unroll posters. Inspect for damage or bugs. Now is a good time to fix tears, use archival document repair tape (I use Lineco brand)
Lay down on clean smooth surface, have ready a large clean flat sheet of material to cover (I use masonite). I usually position with curled ends downward, not like this photo, but it really doesn't matter.
Evenly spray water on one side of "buffer sheet," immediately place wet side against another sheet. This is your "water sandwich." Place these between sets of posters; usually every 6 to 8 works fine. Stack 'em up, but not more than 15-20 or so.
Cover with clean flat sheet of material and weigh down evenly - books, diving weights, motorcycle brake disks...

Wait several days and inspect; if still curled repeat the process.