A dog's nose is always "ON".
A dog is basically an organic battery... needs food, water, warmth and air to operate... and its limited to smells for the most part.
A tricorded has ridiculously diverse scanning capabilities down to the subatomic (well surpassing anything a dog could do) and can last arguably much longer (we've seen evidence that Tricorders can easily outlast a biological organism for example in terms of resources).
The only potential option where use of biological organisms would be beneficial is in environments where technology refuses to work (such as those with dampening fields - but as we saw, even those can be overcome with some clever thinking and modifications to pre-existing technology).
So, no. The Cetacean ops on starships might be different, but even in those cases, I'd find them supurflous in the 24th century given what technology is capable of.
My guess is those still exist because Humanity (among others) established a rapport with various animal species (whom have been long ago declared sentient - heck most non-human animals on Earth ARE sentient) that live on Earth and because humans have drastically changed their ways, they now work together on ships and contribute to explore the galaxy and universe at large.
So, from a strictly operational point of view, SF doesn't NEED animals for their senses, as their technology is lightyears ahead of those already, but they do work together with animals it seems in an equal partnership of sorts... and that's why they have Cetacean ops... might also have others operations on board for other animals... but Dolphins might have been on the more cooperative spectrum with humans so they might be the only (or one of the few) Earth based animals in space on ships as part of Starfleet.