(1) If the rejection is at resume stage, to be honest in my perspective from an ex-recruiter experience, it is quite diffficult to get feedback from HR. HR doesn't do the resume screening (at least for BCG SEA), they pass the resumes collected to the Senior Consultant / Project Manager who are in the recruiting team (as their ECA duty to the firm) to do the screening. Each week these screeners can get > 200 resumes for each of them to go through. It is going to be challenging if they have to provide feedback for the resumes rejected. It could also take time if it goes via HR then HR to reroute the request. HR might not encourage it either (despite the official answers) - what if all the resume rejections come back for feedback? there is no capacity for that at all.
The best way is to ask someone in consulting to take a look at the resume and provide feedback.
(2) If the rejection happens after interview, for sure reach out to ask feedback. For each interview, the interviewer needs to fill a form of evaluation pretty thoroughly, and it can be easily retreaved of what are the strength and improvement areas. Usually the intervierwer would call to tell you the results and share the feedback.