So in Twilight, it’s necessary to burn the vampire to kill them completely. What if only the body was burned and not the head? Would the head continue to survive? Would it just be stuck like a statue seeing everything and not be able to do anything?
Unfortunately, there is insufficient canonical information in the books to give a definitive answer.
The books make it pretty clear that, short of burning, a vampire would eventually regenerate. The most reasonable assumption would be that the head would regenerate a body, but the vampire would probably suffer from excruciating thirst as that happened.
In the books, Edward makes a point that while vampires are very nearly unchanging, there are special exceptions - such as falling in love. It was mentioned that one of the Volturi had nearly been killed by werewolves (children of the moon, true werewolves, and not to be confused with the Quileute) and so has developed a pathological hatred of werewolves. It seems to me that an experience like this would also cause changes in personality.
I think that there would be a credible chance that the vampire might be driven mad with thirst and might not recover, or might not fully recover. In which case, they could endanger the secret of the vampire's existence and the Volturi would send someone to take care of the problem. Other possibilities include catatonia (except when thirsty?), foggy and confused behavior (Drusilla in some episodes of Buffy), or an extreme hatred of the person who left them as a disembodied head (or a group to which that person belonged).