It's impossible to speculate with complete accuracy, but I think we could all agree on some basic generalizations:
It would open up the possibility of a shift of dominance away from the United States. Though much open source software originated in the US, there are no artificial controls to prevent people in other countries using and improving upon that software, and in fact obtaining de-facto dominance of any particular niche.
Some companies' product lines are sure to suffer, and under-diversified companies might go out of business altogether. Intel and Linux delivered a 1-2 punch to companies like SGI and Sun; Oracle may face reduced profits in the future as open source databases like Postgres and MySQL attack their low-end market and creep up; Microsoft has already seen Linux and Apache prevent it from easy domination in the low-to-mid-end web server market.
It's possible that we could see a fundamental shift in the software industry away from predominantly earning money from licensing fees into making money from support contracts, automated update and maintenance services, and consulting services. Some major software companies, like IBM, Oracle, and SAP already make a large proportion, if not the majority of their software-related money from services and support contracts, and that has been the case long before open source came on the scene.
If it ever got to the point that software users came to expect that most software would be free of charge, it would become difficult for an individual or small company to make any money by creating and licensing software. This is already shown itself to be true in the Linux subculture. So much of the software for Linux is free that there isn't much of a market in shareware for Linux, as there is for the Windows and Mac platforms. FTP clients, utilities and other small, purpose specific software that would cost $10-20 (shareware) on Windows is generally available free of charge for Linux, and generally included outright on your typical Linux distribution. Individuals or small firms that might have an interest in distributing a shareware app on Linux probably just don't bother.
So it's likely that if open source software became more widespread there would be some negative impact on the economy. Some companies that are currently in business might be forced to change focus, survive with slimmer profit margins, or even go under as a result. Some companies that might have existed otherwise will never come to be at all. And individual countries, like the United States, might see their dominance in the software industry wane as the market is opened up to other, probably poorer, countries, like India and China.
So in the U.S., software company profits are likely to decline, and some jobs will probably be lost. There will be a negative economic effect from open source software.
Plenty of people have made similar claims, with the most vulnerable of the software firms, industry associations, and their paid mouthpieces being the most vocal. And they do have a point. The problem is, as is the case with any fact that's promoted by an aggrieved party, that there's another side to the story that's conveniently ignored.