People (and this includes even lawyers) are urged to avoid the phrase "burden of proof." If you do some research on the point, you will find that this simple phrase has been used to cover a multitude of different -- often materially different -- concepts, and as a result its use in any form has to be considered unacceptably imprecise.
Herewith a short primer, obviously and necessarily simplified:
In a criminal proceeding, the ultimate burden of persuasion on all elements of any offense, including the non-existance of any exemption, is on the Commonwealth.
Separately, the burden of going forward (i.e., to warrant, but not compel, a favorable finding) is on the Commonwealth on all elements of a stated offense.
However, the burden of going forward with enough evidence to raise an issue is on the defendant with respect to any exception, exemption, or exclusion from a general rule, or any license to do what is criminal without the license.
For instance, if the statute says, "Whoever carries is shall be punished . . . ." and then elsewhere it says "The foregoing shall not apply to anyone who has a license," then all the Commonwealth has to offer by way of going forward is that defendant carried. Burden at this point is on the defendant to introduce minimially sufficient evidence that he was licensed. If that evidence is offered, burden of pursuading the jury that the defendant was not licensed is on the Commonwealth.
In the case ch. 140, sec. 131M, the stated offense is in these terms:
"No person shall sell, offer for sale, transfer or possess an assault weapon or a large capacity feeding device that was not otherwise lawfully possessed on September 13, 1994."
I believe that current law on the burdens "of proof" would hold that the Commonwealth has both the burden of going forward and the burden of persuasion that the "assault weapon" in question "was not otherwise lawfully possesson on September 13, 1994."
Now, that said, I do not disagree with Len's cynicism about how the system may appear to work in practice, but please label it with anything other than the phrase "burden of proof."