Can I reorder my dashboard tiles?

No you cannot. However, the layout of your Dashboard will change based upon the resolution of your screen.

Tiles are organized by the degree of attention needed to push the risk through the sales/placement cycle. 

For example, the Dashboard tile New Web Proposals to Submit, houses new opportunities submitted to you from customers using one of your Web Links; this would be a high priority task to initiate the sales/placement cycle in the platform.