Elmer's Glue is very good at bonding to wood, and it may take several tries to remove it completely.
If the wood is particularly valuable or antique, such as an oak desk or hardwood panel door, consider having a professional woodworker use cabinet scrapers, planer, or sander to remove dried glue, as this may cause less damage.
Amateur efforts can easily damage the wooden surface by using the wrong solvent for the finish, too much force, uneven tool pressure, etc.
If you are dealing with ordinary furniture, and just want to remove unsightly slime so the drawer can be used until the next messy adventure, keep reading.
For childrens' toy slime made with Elmers' glue and Borax, sometimes called "Flubber," the secret is to soak the slime in vinegar. The acidity softens and dissolves the sticky slime.
If the slime is stuck to a vertical surface, you can soak a paper towel in vinegar, and sort of plaster it onto the surface to hold the vinegar in place. Put a plate or placemat underneath, to catch the gob of slime and towel in case they slide off.
Check on it after 5 minutes, to see if it is starting to work. Re-soak and re-apply the paper towel if needed. Leaving the paper towel in one place too long may cause staining of the wood, and/or uneven removal of the slime, with little ridges of residue where the paper towel was wrinkled.
For just plain Elmer's glue, you can soak in the same way with water and/or vinegar. Once the glue starts to soften, you can use a flat tool like a wooden or plastic spatula to gently scrape off as much of the excess glue as possible. The goal is to minimize soaking time, and minimize damage to the wood surface.
Soak the slime until you can physically remove it. Scrape off any excess; if there are little traces of slime in the cracks, soak again or sponge out. Rinse the area with clean water, blot dry with a clean rag, and allow the wood to dry.
Unfortunately, depending on how the wood was finished, you may see some permanent staining or cosmetic damage from food coloring in the toy slime, or from the glue itself and the soaking process.
Once the surface is completely dry, you can assess the damage. Try wiping it down with a wood cleaner and preserver, such as Murphy's Oil Soap. Or apply a small amount of a wood preserving oil such as tung oil or linseed oil, or a very small amount of any vegetable oil from the kitchen.
If the damage remains too ugly and obvious, consider sanding, painting, or refinishing the wood as needed.