Our neighbors have an ingress/egress easement over our property to reach a shed. Can they place a dumpster on our property?

I agree with the other responder but will go this far: you say it is an easement for "ingress/egress" -- over your property to the shed. That means they have the right to get in, and get out. Not put stuff on the easement area. Of course leaving something there temporarily from time to time might itself be a breach of the limitations on the easement as you have called it, but certainly leaving a dumpster on it at all times does not seem to fall within the terms of the easement. So if the easement is in writing someplace -- either on a subdivision plat, or was recorded on its own, or came into existence through some other means, every easement has its limitations, or purposes, and if the "dominant" user (the person in whose benefit the easement runs) abuses the rights given, it can be stopped, first by asking them, and then warning them, and finally if you want to really go full bore, an injunction lawsuit. But before taking any of these steps take the easement document (hopefully) to an attorney to verify your thinking, or find some way of making the situation work for both of you.

Really depends on the language of the easement document (and assuming it was recorded and is truly a valid and enforceable easement). A well drafted easement should limit the uses of the easement area; restrict either party's damage to the easement area, repair, maintenance, etc. Depending on where you are in Illinois, some municipalities also have zoning restrictions on keeping a permanent dumpster on the property. If you need to see a copy of the easement, go to your local recorder's office and you should be able to pull up any document affecting your property, including the subject easement.