When is the best timing for sending an exit survey to an employee who is intending to leave the organization? Is it best to do it before or after they leave the organization? If you know of any related research, it would be very useful.

According to me, Exit interviews should be done before they leave and post leaving survey should be shared for data bank.

I agree with Miren.  Exit surveys are great from a data perspective but you may learn more from an exit interview.  In my career, I've always conducted them prior to the employee leaving though I have heard of other companies conducting them after the employee leaves.  My only concern with conducting them after is the liklihood the employee responds to the company's phone calls/emails.  

Based on practice, by default, it is best to have an exit interview as surveys tend to be less personal. If you have the bandwidth, exit interview is best on the last day of the outgoing employee, when turnover is done for corporate tools and transition/hand-over to employee taking on outgoing employee's load. 

HR should do the interview (not the immediate supervisor) or, if you have a rotating core of exit interviewers (specific leaders or middle managers not related to the team or unit where employee used to work) to ensure confidentiality and reduce bias in documenting the exit interview, that would be best. 

Templated exit interviews help ensure consistency of data. Templates would be most helpful to highlight what they appreciated about the company, what could have been better, and suggestions to the company (aside from validating reason for leaving). Such a format helps prepare employees to exit feeling good about your organization. Even at exit point, outgoing employees should feel good about your company or their decision to leave or even just to come to terms as to why they are being terminated (if it's involuntary).