No. You don't return your licence when you leave. Like many other things in QR it's just another example of inefficiency. This requirement should have been removed from the form years ago.
On a more serious note, the Qatar Government has taken to charging you 500 riyals to CANCEL your Multiple Exit Visa! You must pay this before they'll let you out of the country! This action is nothing but extortion as the visa is of absolutely no use once you leave anyway.
My advice therefore is that you do NOT bother with one unless Qatar Airways decides to buy it for you. (On principle, if Flight Ops wants you to have one, they should pay for it.)
GB