Seeing as Clara has been in the Tardis - which automatically gives people(/living organisms) the ability to understand all languages - why doesn't she understand Dinosaur?
The specific limits of the TARDIS translation matrix have never been explicitly detailed (rendering it a slight case of "new rules as the plot demands," or it could always just be the Doctor talking nonsense, which would not be unusual), but it would seem that the translation matrix appears to draw the line at structurally "crude" languages, which is why the Doctor has also had to translate for babies ("A Good Man Goes to War", "Closing Time"), horses ("A Town Called Mercy"), and a Minotaur ("The God Complex"). Since none of them spoke in such a way that could be converted into a clear, formal language, the Doctor, who has some telepathic ability as well as just a general prowess for reading people and/or situations, had to cover for all of the aforementioned beings, as well as the dinosaur in "Deep Breath".
In the story The Christmas Invasion it was implied that the translation circuit was linked to the Doctor being conscious. In that story the Doctor has recently regenerated and was unconscious, thereby the translation circuit didn't work.
In Deep Breath the Doctor was also recently regenerated, and has not yet recuperated from the trauma, so it's safe to assume that the translation circuit was still not working properly when the dinosaur was talking. (The Doctor probably didn't use the TARDIS to translate the reptile's speech, or the circuit was only partially working and could only translate to the Doctor and not to anyone else)
Once the Doctor regained his strength the dinosaur was already dead, so we wouldn't know whether the TARDIS could translate its words or not.