Whenever someone in Star Trek creates food or drink in the replicator it is generated in the bowl, plate, or cup. It seems like after a few months they would have piles of used dishes from replicator food while making more for every meal. What do they do with all the dishes?
They put them back into the replicator, where the replicator "beams" it away again, i.e. disintegrates it and puts the matter back into storage for future plates.
There is a DS9 episode, where Ben Sisko complains about Jake not putting the dirty dishes back into the replicator.
As you can see below, they put their dirty dishes and cups into the replicator which then reverses the replication process (to allow the material to be used later);
From DS9 : Hard Time
KEIKO : Make sure to put your plate in the replicator.
MOLLY : Okay.
Molly picks up her plate and puts it in the replicator. It disappears...
and DS9 : The Ascent
SISKO : I'm afraid Jake's going through a phase of his own.
I don't know what's so difficult about putting a dirty plate back in the replicator.
There's also a note in the Star Trek TNG: Technical Manual that some replicated materials (water and clothing) are recovered through mechanical and chemical means whereas food is most often returned to the feedstock pile due to limits on resupplies while in space:
Material that cannot be directly recycled by mechanical or chemical means is stored for matter synthesis recycling. This is accomplished by molecular matrix replicators that actually dematerialize the waste materials and rematerialize them in the form of desired objects or materials stored in computer memory. While this process provides an enormous variety of useful items, it is very energy intensive and many everyday consumables (such as water and clothing) are recycled by less energy intensive mechanical or chemical means. Certain types of consumables (such as foodstuffs) are routinely recycled using matter replication because this results in a considerable savings of stored raw material
If you watch carefully, in some episodes they put these items back into the replicator where they are then broken back down into energy and re-absorbed. This is, after all, just a branch of the transporter technology.