Do animals experience osmosis?

Osmosis does occur in animals.  Osmosis is the movement of water across a semi-permeable membrane from areas of high water concentration to areas of lower water concentration.  Osmosis will stop when equilibrium has been reached. For example, if side "A" of a membrane is 90% water and 10% salt, and side "B" of the membrane is 50% water and 50% salt, then water will move across the membrane from side "A" to side "B." 
In terms off where in an animal's body will osmosis occur, it occurs throughout the cells.  I'll use humans to provide a more specific example.  Osmosis plays an important role in the digestive system.  Osmosis definitely occurs in the large intestine because the large intestine's job is to absorb water and salt.  The water is passed from the large intestine to the blood by osmosis because the blood has a "higher concentration of particles than the large intestine."  That means the water concentration is lower in the blood than it is in the large intestine; therefore, osmosis will move water into the blood.   
https://education.seattlepi.com/osmosis-occur-digestive-system-4874.html

https://peer.tamu.edu//curriculum_modules/OrganSystems/module_2/whatweknow4.htm