The NYC Department of Health has released a data set showing the raw number of positive COVID-19 tests for each zip code, along with the total number of tests, on their Github page. We've mapped the zip codes by total positive cases, but you should note that total positive cases is correlated with the number of tests run, so this data may not be a reliable indicator of which neighborhoods have the highest infection numbers, given that all neighborhoods have not received the same amount of testing.
Furthermore, zip codes have unequal populations, so we will need to normalize these numbers to get a per-capita rate that better reflects the situation on the ground.
And here is a map showing positive COVID-19 cases per 100K population by zip code, which will give you a more accurate idea of which neighborhoods have a lot of positives. For reference, Lombardy—the northern region at the center of the outbreak in Italy—would be a 450 here today.
For more NYC-centric data on the coronavirus, you can find our latest charts here.