One common horror story is dealing with difficult landlords. For example, a landlord may not want to make necessary repairs but still expect the leasing agent to find tenants. Another is tenants who damage the property severely. I knew an agent whose tenant painted the walls in crazy colors without permission and then moved out, leaving the agent to deal with the cost of repainting.
Probably tenants who skip out on rent. A leasing agent might think they've found a reliable tenant, but then the tenant stops paying. And when the agent tries to contact them, they've vanished. Also, there are cases where the property is misrepresented by previous tenants during showings. Like they leave it dirty and smelly on purpose so potential new tenants are put off. This makes the leasing agent's job much harder as they have to clean up and make it presentable again quickly.