Here's what the Merriam-Webster online thesaurus suggests (http://www.merriam-webster.com/thesaurus/door).
Synonyms gate, hatch, portalMost of the related words contain door, what we're trying to eliminate. Wicket has too many connections to croquet, and the others are laughable substitutes. As for the synonyms, gate doesn't work for obvious reasons. No one refers to the cabinet doors in their kitchen as gates. Hatch has too many nautical or spaceship associations. I suppose a dotty, old ship captain might use that term but I'm writing a narrative summary and not the speech of a dotty, old ship captain. So, we're left with portal. To me, portal has a grandiose sound to it. It describes a gateway, something that takes you from one type of place to another. A person jumping dimensions would go through a portal. You might use it in reference to the entrance to some building or place with otherworldly connotations, such as a church or a cave. As a joke, someone might say they're passing through the portal to their room, which suggests their room is drastically different, another world or dimension where laws of light and neatness do not apply.
Related Words double door, Dutch door, French door, lattice, portcullis, postern, revolving door, storm door, trapdoor, wicket
As you might guess, door comes to us from Middle English, a merging of the Old English words dor, meaning large door or gate, and duru, meaning door, gate, or wicket. Too bad the distinction didn't persist through the ages, but usage has a way of eroding sharp distinctions in meaning. Portal derives from Old French portal, meaning gate, and the Medieval Latin portale, meaning city gate or porch, which comes from the Latin porta.
Without a proper synonym, I had to do some rewriting to eliminate the word door by suggesting it rather than explicitly stating it. My editor was right. The paragraph reads better now, but the next time you look for a synonym for door, be prepared to bang your head against a door.
Photo Credit: Cattedrale di Santa Maria Assunta, Orbetello. By Nicolò Mumseci [Public domain], via Wikimedia Commons.