Yes, it's possible to protect paper against grease and make it cleanable. I've protected inkjet photo prints on regular paper for hanging in a kitchen, and they were cleanable.
What I used
The coating needed to be smooth and clear for that purpose. I used various clear, solvent-based spray finishes (clear enamel, polyurethane, etc.). Some were better than others as far as not having any color cast and no bubbles (not a factor for this question), but they all worked as far as being cleanable, and none of them yellowed with age.
To be cleanable, it should be solvent-based (look for "aromatic hydrocarbons" on the label), rather than a low-odor, water-based finish. The latter can redissolve when you clean it. I gave the pictures many coats so it saturated the paper and then had a thick surface coat. It resembled lamination.
My situation was a little different, though. The paper wasn't hanging right next to the stove. There's a difference between a regular assault of heavy quantities of hot grease, and cold microscopic droplets that collect over time. The difference in grease quantity also affects how often the paper needs to stand up to cleaning.
Grease and oil break down many plastics. So even if you can clean the surface, the surface, itself, is being damaged. The clear coat used for automotive finishes may be a good fit for this. That needs to stand up to actual oil and other chemicals that a car finish is exposed to.
There's an alternative that would probably work for an application like this question, but it wasn't suitable for photographs. I've used it for other purposes, though. It's clear silicone paint. You can make it by diluting clear silicone adhesive or caulk with paint thinner until it's paint consistency. You can apply it with a brush or make it the right viscosity for a paint sprayer (but you need to clean the sprayer as soon as you're done because the silicone residue won't dissolve in anything once it dries and cures).
Give the paper a good coating. Silicone isn't affected by oil or grease, or grease cleaners, and the coating sticks well to paper. I've only cleaned such surfaces a few times. It cleaned easily and the coating didn't come off. But I don't know how it would hold up to regular cleaning.
If you're just starting the project, use baking parchment instead of regular paper. That is silicone impregnated paper, so the result shouldn't need any treatment or coating. (But I've never tested it to see how well it would perform.)
A superhydrophobic coating like NeverWet might be a good solution. In typical use, they may not hold up long, but that's due to things like constant flexing, exposure outdoors to mechanical assaults like the weather and strong UV light, etc. Those things can degrade the surface that the coating is bonded to. A static object indoors isn't exposed to those kinds of things.
The coating works in a different way. The grease has nothing to attach to. If it doesn't fall off, you can lightly wipe it off; it doesn't need a cleaning process. You could also periodically wipe it clean, then give it another coat.
Your paper isn't flat, which creates challenges both for coating and cleaning it.
Something like 3D origami projections have paper partially covered by folds, and angled surfaces everywhere. That makes it hard to reliably apply a coating in every nook and cranny. There is a good chance of areas not getting well-treated. They will become visible when grease settles there and changes the paper color (and nothing will fix that).
Cleaning will be difficult because you can't apply much pressure, and getting at challenging areas with cleaning tools will be even harder than getting at them with a paint sprayer.
These considerations are different from protecting a flat surface like a picture. The best solution for this situation may be saturating the paper well with a product like NeverWet. That will minimize grease sticking in the first place, and is cleanable with very little pressure.
As an alternative, I'd be tempted to try a two-step solution since failure to protect the paper would result in needing to discard the project. I would first use one of the clear spray coatings, after verifying that the product is cleanable with paint thinner after it is dry. Once it's dry, I would spray silicone paint, making sure folds and seams are well bridged. The paint thinner in that will help it bond to the previous coating. The two steps would ensure complete coverage, and the silicone top layer is cleanable without needing to apply much pressure.