We know he was removed from office by Odoacer in 476 and pensioned off to relatives in Campania. But what happened to him after that? Do we have any sense of what his life was like, whether any opposition formed around him, when he died?
Sadly it is impossible to know what happened to Romulus. The nature of his deposition itself is disputed. The rather generous account you mentioned, that he was pensioned off by Odoacer, is found in the Anonymus Valesianus, a propaganda pamphlet composed after Justinian's reconquest of Italy - which may have perhaps affected its account of the disappearance of imperial rule in the fifth century. Certainly, we know that the anonymous author gave Romulus an implausible reign of 'ten years' rather than less than one year, so we should be careful when following their narrative too faithfully.
Other authors are rather more negative about the fate of Romulus. Marcellinus Comes, the first author to suggest that 476 was a decisive moment, wrote in Constantinople c.518 that Romulus was exiled to Campania, without any mention of a pension. Jordanes, another Constantinopolitan author writing in the 550s, similarly described how Romulus was condemned to exile. Then again, both Marcellinus and Jordanes were very much part of the imperial apparatus and they had a vested interest in pointing out how the downfall of Romulus Augustulus was a lamentable disaster - a disaster that was only remedied when their imperial master, Justinian, restored the Roman state with his reconquest of Italy. Given these differences, I'm not so certain that we can even be sure about exactly what happened to the fallen emperor in 476!
The one possible piece of information about Romulus after his deposition comes from an official letter from King Theoderic of the Ostrogoths (and the ruler of Italy) to a certain Romulus in 507 or 511. The letter says very little, only confirming a previous grant to this Romulus and his mother. Nor can we be sure that this Romulus is Romulus Augustulus, as the letter does not give us any contextual information to help us to identify this figure. Romulus is a rare name in late antiquity, but between the fourth and sixth centuries there were seven other people named Romulus (and I am sure that there must be many more, I only checked one prosopographical resource), so we should not automatically presume that this person must be Romulus.
TL; DR - we don't know.