I could not find any official sources to verify the following suggestions on how to eliminate the cabinet squeak without damaging your posters, so the following is suggestion only. All suggestions come from colleagues who work with/have knowledge of archival materials (thanks to Elizabeth, Tina, Simeon, David and Da).

First examine why the drawers are squeaking. If they are made of wood humidity may be causing them to expand and squeak. In this case an air conditioner can control humidity in the room and will eliminate squeak. If the drawers are steel examine the bolts that hold the drawers together - they may simply need to be tightened or cleaned and refastened.

You may also try the follow products to eliminate "squeak-age":

1) use beeswax or candlewax
2) crushed graphite
3) vaseline or baby oil (not vegetable oil)
4) A metal cleaner/lubricant purchased from an specialised supply company

If anyone has a source to back up these suggestions or wants to add their own suggestion post a comment to this answer.