It'll make a great meat-shield, and provide excellent cover to hide behind. Often whether you want it to or not.
If you are unwilling to change your mount, then the most reasonable answer is "Change yourself instead."
While dragging a horse into a dungeon can be done, it's probably not a good idea.
At best the dungeon will smell like monsters, and the horse won't like that, making it more difficult to control. This is less true with a war-trained animal, but then the animal will be expecting a fight and will be more difficult to control in that direction.
And then, assuming you can get it to go into the place that smells so questionably, you have to ask the next question.
Will there be enough room in any given direction? A horse can't really crouch and move. If the ceiling is low, you won't be able to ride without scraping on it. If the ceiling is lower, the horse simply won't be able to progress. This is equally true for width; if you are brushing your shoulders against the walls walking straight in, the horse is getting squeezed. If you have to turn sideways to pass, the horse simply isn't going.
Another answer suggested "Only take on dungeons with large corridors" but what do you do when you explore deeper into the dungeon and discover that only the "ground floor" has large corridors, and everything else is much smaller?
In AD&D the default was a 10' square, which would provide enough room. Not a lot, but enough. D&D (3E and later) changed that to the 5' square, meaning your horse now occupies a 2x2 space on the map.
Now that your horse is nervous, and cramped in a tight space, you can deal with the party's "marching order".
Horses are animals, and react by instinct not reasoning.
One of those instincts is to kick when something "sneaks up" on it. They can't see behind themselves any better than we can, so when a party member walks up behind your twitchy horse, they risk a sudden kick in the face.
(Another instinct is rearing to attack an enemy. Won't that be fun for the rider in a low-ceilinged area?)
Plus, if the horse gets stuck and can't pass, well so does any of the party in line behind it.
You can put the horse (and rider) at the back of the line, which isn't a bad place for the archer to be either, but the horse can't tell you that it hears something coming up behind you, and it can't turn around to hold the monsters off for a couple of rounds while the lineup reorganizes itself.
I won't even consider this to be a complete list of potential problems. (Leg injuries after the magical healing is exhausted?)
Trying to bring a horse into a dungeon is a liability, not a benefit. Even when you can afford the resources to make it work, it still requires that those resources be dedicated to that task, instead of being used somewhere else.