The landlord is responsible for extermination, but as the tenants introduced the fleas into their unit the landlord can likely pass the cost along to the tenant. The shared yard is an interesting wrinkle, but depending on the nature of the yard it may not be feasible to eliminate fleas, and it will be difficult for the tenant to prove both that the fleas in the yard were introduced by somebody else's animals and that the yard was the source of the fleas in their unit.

If the tenants are correct about the source of the fleas, even if the landlord sprays the yard, the problem is likely to recur within weeks as the animals that introduced fleas to the yard return and re-infest the yard.