What's up with Tobiah's dog? Either Tobiah or Archangel Raphael (or maybe both?) has some sort of relationship with a dog, as shown in Tobit 6:2 and Tobit 11:4.  The dog goes out with them and comes back with them and isn't otherwise mentioned.  Does the dog have any special significance?  Given Tobit 2:13-14 it seems pretty odd for Tobiah to have picked up a dog somewhere in the past.
Those studying and commenting on Tobit have long been intrigued and puzzled by the presence of Tobit's "canine companion". This answer is based on Carey Moore's work, Tobit: A New Translation with Introduction and Commentary (Anchor Bible 40; Doubleday, 1996). He signals the measure of bemusement this creature brings as he begins his comment on the 11:4 occurrence (p. 261):


  That elusive dog again!


He offers most of his discussion about this canine character at 6:2 (pp. 197-8). Condensing and simplifying, with a few notes of my own, the main options canvassed by scholars are these:


An old scribal mistake, which turned ha-leb ("the heart") into keleb ("dog") -- but this is an old suggestion which has not been taken up.
An "echo" of the "Tale of Ahiqar", counselling careful speech, since "the tail of a dog gives him bread, but his mouth gets him blows". This seems like a fairly dim and distant allusion (if it is one) to me (although there is an "Ahiqar" character in the story, cf. 1:21-22; 2:10; 11:18; 14:10).
There are several suggestions which cluster around the role of the dog in Persian/Zoroastrian culture and ritual; cf. Mary Boyce, A History of Zoroastrianism: 1. The Early Period (Brill, 1996), pp. 302-3 ("It is possible to find rational and historical grounds for the Zoroastrian regard for the dog."); and F. Kotwal and J.K. Choksy, "To Praise the Souls of the Deceased and the Immortal Spirits of the Righteous Ones", in M. Stausberg (ed), Zoroastrian Rituals in Context (Brill, 2004), p. 399. But while this might add an element of verisimilitude to the story, there is no ritual role for this particular dog.
Moore's own view is that Tobit draws on an older "secular" folk tale, and "the safest conclusion is that the dog, which is the only gratuitous piece of description in the story, was originally part of the secular folktale used by the narrator" (p. 198). At the 11:4 context, Moore concedes that the dog might have played a more integral role in that "original" tale, and that what remains is fairly vestigial.


Important to note for OP's form of the question, though, is that the dog has no particular relationship with the archangel Raphael, but appears to be part of the Tobit household (or perhaps a friendly "neighbourhood" dog): the two explicit mentions of the dog marking the beginning (6:2) and end (11:4) of the journey.