There is no profit in a free market system for open-sourced innovations. With open-source innovations, proprietary information is completely public and can be shared with anyone. If all innovations are open-sourced, an entirely new economic system will need to be developed because capitalism will no longer be a viable way to make money.
No, innovations should not be open sourced, because it would lead to a bootlegged industry. A person who invents something great should be able to make a profit from it. Otherwise, people will not put the time into inventing things. Open sourcing should only be for those who are willing.
The question is a bit confusing because open source generally refers to computer programs but innovations cover a wide array of technologies many of which have nothing to do with computers. I am assuming you are asking about all innovations and whether the underlying methods should be revealed, in which case it should not. Case in point was white-out, which was forced to reveal its ingredients to obtain a patent but was subsequently copied by hundreds of manufacturers who simply changed the ratios a little bit. Of course, that is all done on computer these days so perhaps the question was more accurate than I thought at first.
Intellectual property is one of the great motivators for hard work and discovery. Without this incentive companies and individuals wouldn't work as hard as they do and they wouldn't invest the considerable resources that go into innovation. It would be a disaster if these incentives weren't available. It is true that everyone would have access to new discoveries, but there would be much fewer of these discoveries.