Last time it happened for one of our sites it was ~1500 visits or so. I'm sure it various depending on content/interest/day/etc.
Though I'm not sure why you'd be so keen on the actual numbers. This isn't the sort of site where you're really looking to get pageviews-- it's more about the great feedback and comments really.
Going through my server logs counting lines which were (a) within 2 days of my submitting a link here, and (b) matched /ycombinator/i, I get the following:
Of course, this doesn't count traffic from sites like icombinator.net which "re-skin" HN, nor does it count traffic resulting from people seeing links here and posting them elsewhere.
Though I'm not sure why you'd be so keen on the actual numbers. This isn't the sort of site where you're really looking to get pageviews-- it's more about the great feedback and comments really.