In a pitcher or large cocktail shaker, combine the lime juice, sweetened condensed milk, water, tequila, and orange juice. It is important to combine the ingredients in a container that has a spout because it will make it much easier to pour the mix into cups.
Line out the Dixie cups on a baking sheet, so it will be easy to transport them to the freezer. Fill each Dixie cup 90% of the way with the tequila mixture.
Slice the limes into 1/4 inch thick rounds. Push a craft spoon through the center of each lime round, so that half of the stick pokes out from each side. Top the mixture with the lime sticks. Make sure the lime is resting directly on top of the tequila mixture.
Freeze for at least 6 hours, or overnight. When ready to remove the popsicles from the Dixie cups, simply snip a small cut in the rim of each cup and peel them away from the popsicle. Place the popsicles onto a dish with crushed ice to keep them from melting too quickly, Sprinkle the tops generously with kosher salt, and serve.