Yes! Glass recycling facilities will clean, sort, and crush glass, and then melt it at 1600 degrees, well above the flash point of a lime wedge. Glass bottles and jars are not reused, but rather recycled, so it's not a situation where the lime has to come out in order to refill the bottle.
We're asked to make sure items are clean and dry before recycling them for the following reasons:
1) Cross-contamination. Most communities in the U.S. have single stream, not dual stream, recycling. As this infographic demonstrates, in dual stream recycling paper/fiber items like newpaper, magazines, and cardboard go in one cart, and everything else goes in another.
Unlike metal, glass, or plastic items in your bin, paper is typically processed at temperatures below the boiling point (212˚ Fahrenheit). In fact, anything above 250˚(F) can cause damage to the paper fibers being recycled. Because of this, the paper recycling process won’t burn off food residue in the way that other materials-recycling processes might.
In dual stream recycling, paper is in a completely separate cart so contamination doesn't happen. But with single stream recycling, if something is half-full of food or water, it can ruin the quality of any cardboard or paper in the bin. And water shortens and weakens paper fiber, making it not valuable for recycling.
Glass, plastic, and metal are less sensitive to contamination from food residue because of the heat used in the recycling process of these materials, but that's not the case with paper. These other materials don't need to be 'dishwasher' clean - but they need to be clean enough not to contaminate the paper and cardboard in single-stream recycling carts.
2) Empty, clean, and dry recyclables help reduce or prevent odors, mold, and pests such as insects, rats, and raccoons, both at home and at recycling facilities.
Want to learn more about how glass recycling actually works? Check out this illustrated description from Momentum Recycling: https://utah.momentumrecycling.com/glass-recycling-process/