Edit: Perhaps the answer to this question lies in disclosure – how would the sniffer dog make their statement? This article from the Daily Mail might shed some light on that.
My opinion was sought on this question several years ago. “Of course not,” I replied without hesitation. “Even if it could remember the oath, how is it going to hold the Bible?”
After the decision of the New South Wales Court of Criminal Appeal in Muldoon v R; Carter v R  NSWCCA 315, the question might be worthy of more serious consideration. In Muldoon, three accused were apprehended by a tracker dog being exercised by its off-duty handler. The animal followed the men from a burglary to some adjoining bushland.
As the dog couldn’t give evidence of its own observations, the Crown sought to lead evidence from the handler about the dog’s training, abilities and actions. The handler testified about what he saw the animal do (which was clearly admissible) and also gave expert evidence about why the animal behaved and reacted as it did.
Counsel for the accused argued that evidence was inadmissible under s 137 of the Evidence Act 1995 (NSW) because the prejudicial effect of the evidence outweighed its probative value. The defence also relied on underwhelming scores the animal achieved in previous training exercises to argue the ‘evidence of the dog’ was inherently unreliable.
The NSW Court of Appeal rejected the arguments and ruled the trial judge was right to admit the evidence. The Court differed from the less accepting approach in the Victorian case of R v Joe Saccu (unreported, Victorian Court of Criminal Appeal, 13 February 1980). Further discussion of that case, as well as a thorough discussion of the topic internationally, is in Lawbook Online’s Expert Evidence at [13.10.01] – [13.10.260].