There are two problems here that need to be addressed. The first was that Carthage was a Merchant Empire first and foremost, and therefore every decision that they made was motivated by money. While Carthage was nominally ruled by the Senate, in reality power was held de facto by a group of incredibly wealthy oligarchs that had gained most of their wealth through trade. There was a sizable portion of these elites who formed an anti-war party in Carthage, as they viewed another war against the Romans as an unprofitable venture. So there was a large group of very wealthy individuals in Carthage who actively worked against Hannibal garnering support in Carthage proper.

The second problem here is actually with your question, in that there was support coming for Hannibal in Italy, led by his brother Hasdrubal. However, by this time Carthage had lost its naval hegemony in the First Punic War. No longer able to guarantee a landing in southern Italy, both Hannibal and Hasdrubal were forced to travel all the way through Hispania (modern-day Spain) and over the Alps in order to get into Italy. Hasdrubal was actually more successful in getting into Italy than his brother was (Hannibal's army was hit by a storm in the Alps), but as soon as he got to Italy he was met by the Roman army and defeated decisively at the Battle of the Metaurus (Livy 27.49) where he died. Carthage had no interest in raising the money to send another force all the way through the Alps to attempt to relieve Hannibal. Besides, the only other member of the Barcids (Hannibal's family) who would strongly push to relive Hannibal was already in Italy fighting with his brother.

Similarly, it is not quite correct to say the Hannibal didn't do real lasting damage to Italy, or Rome in particular. The Fabian Strategy employed to combat Hannibal was heavy attrition warfare which involved destroying much of the land so that Hannibal couldn't use it to resupply. Hannibal also did heavy damage to the adult male populations of many cities in Italy, both Rome and the allies. Livy tells us that roughly 50,000 Roman and allied men died at the Battle of Cannae. Even if that number is slightly embellished (Polybius actually gives a higher one, and Livy's is generally considered correct), that is still a large portion of the population of Italy. That was one battle, and Hannibal was immensely successful for months.

Hannibal was also forced home shortly thereafter because Scipio had landed a sizable army in the Carthaginian homeland. This caused Hannibal to give up his conquest of Italy and return home. By the time Scipio and Hannibal met at the Battle of Zama, the Roman Army had figured out how to deal with some of Hannibal's tactics, and Hannibal's depleted army was defeated.