Q:

Intentionally Breaching a Contract

Hypothetical Scenario:
Suppose I'm a landlord in a contract with a tenant, and I may only inspect the premises if it has been three months since the last inspection.
Let's also suppose however that the contract states that the tenant must clean away all mould from the premises, and if mould is found during an inspection, the tenant can be asked to clean it.
Now suppose I arrange an inspection of the premises one month after the last, in breach of the contract.  
Questions:

While I have breached the contract by inspecting the premises a second time in three months, if I have not caused damages to the tenant, could there be any legal repercussions?
Suppose in the second inspection I find mould in the premises and issue the tenant a breach of contract form (for not cleaning the mould).  Could the tenant claim the costs of cleaning the mould as "damages" for my breach of contract under the "but for" test?

A:

if I have not caused damages to the tenant, could there be any legal
  repercussions?

If by "arrang[ing] an inspection" you mean that landlord and tenant agreed to have the premature inspection, then the tenant has no cause of action against the landlord. I am not knowledgeable of Australia legislation, but I highly doubt that the statute prohibits the parties to willfully agree on having that premature inspection.
Note that if the tenant received the notice and did not object to it, then that would constitute tenant's implied agreement to have the extra inspection to the extent that the tenant knew or reasonably should have known about it.
If the tenant objects to the landlord's notice, then the landlord would have to comply with the contract or proceed in accordance with the Residential Tenancies Act 1997. In particular, see sections 76(d) (entitling the lessor to access the premises if safety reasons in relation to the premises can be justified) and 79(3) (either party "may apply to the tribunal for an order permitting access at a specified time").
(In line with the initial version of my answer, note that section 76(e) allows the parties to willfully agree to have extra inspections)
Also note that section 79(2) requires that the inspection notice be given to the tenant one week in advance.

Could the tenant claim the costs of cleaning the mould as "damages"
  for my breach of contract under the "but for" test?

No. The tenant's duty to remove the mould applies constantly (more precisely: with reasonable or practicable frequency) and irrespective of when the inspections are conducted. 
Likewise, it is reasonable to enforce the tenant's duty from the standpoint that the effects of mould (such as a reduced value of the property) materialize regardless of inspections schedule.

