The Residential Tenancies Act 1997 does not supply a ready answer to this question. It does not say that you are responsible for pest control of the premises, nor does it say the landlord is responsible.
It’s clear that a landlord, or their agent, is obligated to maintain the premises in good repair and in a state fit for you to live in (clause 55 of the Standard Residential Tenancy Terms). They are also required to make repairs when necessary (clause 57) but can this be extended by analogy to cover fumigation for pests?
The Standard Residential Tenancy Terms say that tenants must keep the place clean and not damage the premises (clause 63). If you follow your responsibilities under the Act and don’t cause an infestation of pests, then the landlord will be responsible for pest control as part of their responsibility to maintain the premises.
You will only be responsible for pest control if you have caused an infestation, for example by keeping pets on the premises. If you do your own pest control or have professional pest control carried out it is important to keep receipts and a record of what you have done. If there is a dispute about pest control, this evidence can show that you have taken reasonable steps to reduce any pest control problem. An important factor in this type of situation might be evidence as to why the vermin are there in the first place. Getting a pest-control person around for an obligation-free inspection might be helpful here. A landlord is much more likely to accept the cost of fumigation if he or she is convinced that the presence of the vermin is not a result of the tenant’s conduct.
A final possibility comes from section 71 of the Act, which allows a tenant to apply to the ACT Civil and Administrative Tribunal (ACAT) for a rent reduction. If for example a hive of European wasps renders the backyard unusable, then it might be possible to argue for reduced rent unless or until the problem is fixed or goes away. Section 71(1)(b) only requires that the tenant’s use or enjoyment of the premises has diminished as a result of the loss of the use of all or part of the premises, it does not require that this situation comes about due to some failure by the lessor to fulfil his/her obligations. So a landlord could be ordered to allow the tenant to pay a reduced amount of rent. Note that section 71 allows such a rent reduction to be backdated as well, for example to the date that the problem arose (which might well be the date that the tenant moved in).
As always, keep records such as photos of the pests, copies of quotes or receipts, and put all correspondence in writing (sign, date and keep a copy).