The determination of whether or not to evict is up to the landlord. However, violating a no pets provision would give the landlord a valid basis to serve notice upon the tenant and, if the tenant does not get rid of the pet by the end of the notice period, to commence an eviction case.
The tenant could consider asking the landlord if the landlord will approve a pet, despite the broad language of the lease. While we can't make any promises, some landlords may be amenable to allowing certain pets, but may also be inclined to charge additional deposits or fees (consistent with the laws of their state).

