A new version of this post has been published here:
Is it possible to hide all the zeroes in this table? Yes, and it’s easy!
Select the cells, go to the home ribbon, choose Conditional Formatting => Highlight Cell Rules => Equal to...
Type a zero in the left field and choose Custom Format in the drop-down menu.
Go to the Font tab and choose white colour (or light grey, if you don't want to hide them completely):
And now the table looks nice and clean: