To the best of my knowledge, there is no canon answer to this. It would also potentially change depending on the canon, even if there were a canon answer in one of the various bits of media Beast Boy is in. For example, even if he were shown in Teen Titans GO! to be able to regenerate a lost limb, that wouldn't necessarily carry over to meaning that his comic incarnation could (or even his Teen Titans TV show version).
That said, Beast Boy has shown considerable flexibility in his power. He can change form into something with many times his mass (a whale) or with significantly less mass (an insect). He seems to be able to perfectly mimic most animals (excepting color).
It seems to me that there are two possibilities, based on his commonly displayed abilities, that the writers would have to decide between:
Each of these would pose different difficulties for a writer. If Beast Boy is automatically fully healed by changing form (excepting mental effects like exhaustion) then he becomes an extremely difficult person to physically harm. If Beast Boy retains injuries between forms, what happens when he is injured on a part that doesn't have a corresponding location on his normal form? What happens when he changes into an octopus and gets a tentacle injured?
Finally, we have to consider the role that Beast Boy plays - BB is most often the comic relief or butt monkey. He's not the sort to brood or be dramatic. A character like Raven or Arsenal could lose a limb and it would fit their character - they would be able to talk (to the others or the audience via thought bubbles) about what it means to lose a limb, they could evaluate their self-worth, berate their own weakness, etc. For those characters, it would be a way to explore their roles and convey the messages they're supposed to pass along to the reader. With Beast Boy, it wouldn't work. Comics have tried having the happy-go-lucky goofball turn morose (I'm looking at you, Speedball) and it has universally been a failure.
In short, Beast Boy's powers exact limitations are up to the writer and can vary by media and series. For that reason, this question cannot be given a definitive answer unless and until the problem is presented in an official work. That said, I expect the Beast Boy would never permanently lose a limb - he would not be able to serve his purpose in the narrative if he did. Thus, if he ever does lose a limb in canon (with the exception of time travel/Dark Future/dreams/etc) I expect he will find a way to regain it. It may be explained away as a one-time use of his powers (like when Superman rose from the dead) or as a wish or the result of use of some strange medical device, but he would regain it.