I don't believe there are any modern heirs to the Medici Bank, simply because it never existed independently of the individuals that constituted it: it was a partnership first founded in the late 14th century (and in fact each "branch" of the Bank was also a partnership, headed by sons/brothers/nephews of the Medicis or close allies), and its rise and fall all took place within the span of a century. de Roover's "The Rise and Decline of the Medici Bank, 1397–1494" remains the best source on this.

Among banking historians, I think the Bank of St. George (Banco di San Giorgio) in Genoa, founded in 1407, has recently overtaken the Medici banks in importance: I believe it is the first recorded instance of an incorporated bank (independent of the individuals that founded it), but more importantly it was the first successful prototype of a state deposit bank (and therefore a precursor to institutions like the Bank of England and the Federal Reserve). It lasted a lot longer than the Medicis, until 1805 - and was recently resurrected, so you might count that as the oldest bank we can link to the present time. There's a beautifully put-together book by an Italian historian, Giuseppe Felloni, using the bank's own archives - it's available free on his website.

But maybe the best example is the Banca Monte dei Paschi di Siena, founded in 1472 and operating continuously since then, which would make it about 542 years old this year. You can thank the Italian government that the clock didn't stop ticking at 541: the bank was bailed out last January to the tune of about 4 billion euros. They raised money on the stock market last month and have begun repaying the government (narrowly avoiding de facto nationalization) as of two days ago, so no reason not to count on them making it to 543!