Whether the object (either privilege or a burden) has been taken on voluntarily, earned, gifted or inherited doesn't matter IMO - the idea is that in either case the responsibility _to_ the object lies _on_ the subject. For example - you own a factory with 100 workers. It is a privilege to lead and conduct such a business. At the same time, it is a burden that needs to be carried - the business must be ethical; workers, contractors & partners paid on time, it must be profitable (or at least potentially profitable in the future), etc. 

Kontext

Stacker Saloon

saloon

I don't think so.
A privilege is something that somebody else (usually your master) is giving you as a reward for being a good obedient slave.
A burden is something you take for yourself to "carry", so it is with your own consent and will.