What will happen to a poor country’s economy if it replaces its entire banking system with the Islamic banking system?
I see a lot of research on Islamic banking, but I don't see a lot of studies that speculate on the question framed here. You might be interested in this study where the author claims:


  It would be better if Islamic banks had the opportunity to work as a sole system in an economy. That would provide Islamic banking system to fully utilize its potentials. Studies show that Islamic banks can not operate with its full efficiency level if it operates under a conventional banking framework, their efficiency goes down in a number of dimensions. The deterioration is not because of Islamic bank’s own mechanical deficiencies.  Rather  it  is  the  efficiency-blunt  operations  of  the  conventional  banking  system  that  puts  obstructions to efficient operation of Islamic banks.


However, it doesn't look like these assertions have been proven. For example here is a study that suggests that Islamic finance works best on a small scale as larger-scale Islamic banks seem to have unique instability issues.