The clause is void on signing
Section 14 is one sentence: "A provision in a tenancy agreement prohibiting the presence of animals in or about the residential complex is void." There is no exception for a tenant who agreed to it, and none for a clause negotiated in exchange for something else.
Void means it never had effect. You cannot enforce it, you cannot charge for breaching it, and a tenant who brings home a dog has not broken the tenancy agreement in any way the Board will recognise.
What section 14 does not do is force you to rent to someone with a pet. It governs the tenancy agreement, not tenant selection. Selection is governed by the Human Rights Code, and refusing a service animal is a very different problem from preferring a quiet tenant.
The three grounds that do work
Section 76(1) lets the Board terminate a tenancy over an animal only where it is satisfied the tenant is keeping an animal and one of three things is true.
First, "the past behaviour of an animal of that species has substantially interfered with the reasonable enjoyment of the residential complex for all usual purposes by the landlord or other tenants" — the barking that never stops, the dog that lunges in the corridor.
Second, "the presence of an animal of that species has caused the landlord or another tenant to suffer a serious allergic reaction". Serious, and actually suffered, not anticipated.
Third, "the presence of an animal of that species or breed is inherently dangerous to the safety of the landlord or the other tenants".
Subsections 76(2) and (3) then close the obvious gap: the Board must not evict on the interference or allergy grounds if it is satisfied that the animal this tenant keeps did not cause or contribute to the problem. You cannot evict a quiet cat because a different tenant's dog is the issue.
Two other clauses that are void the same way
Section 15 voids an acceleration clause — a provision making the rest of the term's rent, or a fixed sum, fall due the moment a tenant defaults. If your lease has one, it does nothing.
Section 16 imposes a duty to minimise losses on whoever is claiming: "the person entitled to claim the amount has a duty to take reasonable steps to minimize the person's losses." It applies to landlords and tenants alike, and it is the answer to a landlord who lets damage worsen and then bills for the larger repair.
What to do instead
Delete the no-pet clause. Keeping an unenforceable term in a lease signals to an adjudicator that the rest of the document may not have been read carefully either.
Write what is actually enforceable: the tenant is responsible for undue damage under section 34, and for not substantially interfering with anyone else's reasonable enjoyment. Both cover the real risk without pretending to ban anything.
If a specific animal becomes a problem, the evidence is what decides it — dated complaints from other residents, the notes you made at the time, photographs of damage. A section 76 application is won on a record, not on the clause you wish were enforceable.
A condominium's declaration and rules are made under a different statute and can restrict animals in ways a lease cannot. If you rent out a condo unit, read the declaration before you promise a prospective tenant anything.