What does soffritto do to minestrone? I have read some recipes for minestrone where it calls for making a soffritto before adding water. What would happen if you omitted this step and just added raw onion, carrot and celery to the boiling water, as you would do with the rest of the vegetables?

The vegetables don't need to be browned, hence it appears to me that shallow frying them is not necessary if it is followed by boiling either way.
A soffritto is the Italian cousin of the French mirepoix. Both consist of small cubes of root vegetables and onions. 

The gentle “sweating” in fat enhances the sweetness of the vegetables and brings out the “umami”, an almost meaty flavor. In the onions it also breaks down the sharp pungency. The process will form a flavor base that brings a certain “heartiness” to stews and sauces.

You can skip the step - many soups will use the raw, coarser chopped vegetables - but the results would not have the properties listed above.