Can ongoing economic growth occur without use of additional non-renewable inputs? That is, any additional inputs would only be from renewable resources: all non-renewable resources would be reused, recycled or removed from the economy.
This would fall under the category of endogenous growth models. As dismalscience pointed out innovation is one source of growth, however it is important to distinguish between innovation and technological change. Innovation is sometimes confused with technological change, which is an exogenous process, but is more often associated with endogenous growth.
Endogenous growth theory involves knowledge spillover, learning by doing and more generally increases in human capital as the driver of economic growth. These models were explored extensively by Romer(1986), Lucas(1988) and Rebelo(1991). In essence these models result in growth through the assumption that people get better at doing things the more they do them. Growth is often modeled as an externality in the form of knowledge spillover or through human capital making physical capital more productive. This can result in a violation of the Inada conditions, which differs from the growth models developed by Robert Solow.
I don't think anyone knows how to "make" an economy grow, but there's a simple channel through which economic growth can occur under fixed non-renewable inputs: innovation.
Economic growth does not require increasing use of inputs; if there's profit to be gained through more efficient use of fixed quantities of inputs, or by increasing output of renewable resources, people will have an incentive to figure out how to do so.
Regarding whether fixing a minimum price of a nonrenewable resource is a good idea: possibly, but likely no. Typically, the most efficient ways to reduce the use of a particular input are to either charge the marginal social cost of the use of that input as a tax (allowing quantities to vary freely, which you disallow in your scenario), which incidentally has the effect of setting a minimum price but which requires that you know the marginal social cost of the use of that resource, or to do cap-and-trade, where you fix the quantity available and let market prices determine the most efficient allocation. Setting a minimum price in a cap-and-trade scenario can result in inefficient underuse of the resource.
If you're interested in the theory of the price of nonrenewable resources, you might want to start with Hotelling's Rule.