No you cannot. A spell or ability has to resolve fully before any player gets priority. Once a round of priority has been passed with no action, the topmost object on the stack resolves. Once that object resolves, that's when players will get priority again, starting with the Active Player.
April 10, 2016 1:21 a.m.
If you want to make sure you're not dumping cards you want to keep, you may want to consider running Library of Leng in conjunction with Burning Inquiry. It takes the randomness (sort of) out of the discard and lets you put some, if not all of them back on top of your library.