What are my options for dealing with/getting rid of storage items I do not want/need?
You are responsible to see the items all the way to their final disposition. 

Yes, it's a pain, but you have to do it.  


You can ask the city if they can do a special garbage pickup.
You can call your normal garbage company and ask them about a special pickup. 
You can rent a dumpster short-term and fill that.  
You can ask your movers to dispose of the items in their dumpster, however, if they dump the items down some alley instead, and the government catches you, you will be fined heavily even though you acted in good faith. 
You can hire "get rid of your junk" companies to make it go away; they may try where possible to profit from its sale, however the same warning applies. 
You can put a "free stuff" listing on Craigslist, Freecycle, Nextdoor etc.  -- but if the stuff is not taken, you are still responsible for its disposal.
If you are renting, you can walk away from the stuff.  The landlord will hire movers to remove it, and bill the cost of this to you.  


Taking the items to Goodwill/charity is probably a lose; they are glutted with excess furniture, so they are very picky what they'll take.  

If you abandon the items at a house you are selling, it could interfere with closing, and at least, the buyer could sue you in small claims for the disposal costs.  "Broom clean" is the standard for both sold houses and rentals; look it up. 

My best advice is if you don't want to dispose of it, don't buy it in the first place.