What did Italian dishes consist of before they came to the new world and discovered tomatoes?
There are all kinds of examples of Italian dishes that don't have tomatoes.
If anything, the association between Italian food and tomato is probably more because of Italian-American cuisine in the US. In Italy tomatoes aren't terribly common except on pizza or certain pasta dishes.
So the answer is... basically everything that is eaten in Italy now, minus the one ingredient that happens to be a New World thing.
(Also worth mentioning is that tomatoes didn't catch on as a foodstuff right away in most of Europe, including Italy. Since tomatoes are in the nightshade family, they were thought to be poisonous and generally distrusted as a food source until the 17th century.)
EDIT: I also want to clarify that I'm not asserting that Italian cuisine is otherwise unchanged since the pre-Columbian era, of course! Just that there's no reason to think the presence or absence of tomatoes would have been significant at all. Here's a pretty cool site that features some 14th and 15th century Italian recipes based on research done by an SCA member. It's hard to tell from looking at these recipes how Italian cuisine at that time would have differed from other European styles of cooking, though I notice a lot of recipes featuring oranges and other Mediterranean fruits. That said, I'm wondering how much this specific selection has to do with the source, since someone translating recipes for use at SCA feasts is more likely to choose things that sound appetizing to the modern palate.
(It's also worth noting that a much bigger "what did they eat before New World foods?" question would really be the arrival of corn in the form of polenta.)