Does anyone know if the U.S. sanctions prevent someone in the U.S. from sending a letter or parcel directly to Burma? I can't get a clear answer on the State Department or U.S. Postal Service websites.

No, the problem is the Myanmar government, and the poor mail system, not even sure they have a mail system to speak of... Most things never lake it, or get opened and inspected and never delivered, as they are sensitive to the nature of all communication and materials going In/Out.
Its best to try to have someone deliver it for you, trusting a fellow traveler heading there may be best.
I am not saying it will not get there and delivered, but odds are, not too good....
As with Email, and Internet and cell phones, the government is pretty strict.
When I offer to send them something, my friends in Bagan and Taunggyi always tell me to bring it along the next time I come rather than send it in the mail. Even things we might feel uninteresting to most people will usually get removed from any letter or parcel before it arrives at it's intended destination. I tried to send a friend the earplugs he longed for (his wife snores) and they were removed. Even pictures taken of friends that I sent with a letter were taken and just the letter arrived.

I understand the problems with the Myanmar postal system. However, I want to know if it will even make it out of the U.S. without being returned.

I am not aware of the sanctions effecting mail from the USA to Myanmar...but at least it will get returned to you.

I send mail to Myanmar all the time and it gets there . Do not send cash even if it's by registered mail. I doubt the mail screeners are looking for anything other then money. It cost .90 for a regular airmail letter and around $9.00 for registered from the US. My photos are not confiscated.

The US PO says that we still have sanctions on Vietnam, although I'm not sure if there is any truth to that any more??? I thought they were already lifted and relations were normalized. Still, regardless of what the US PO believes, mail gets to VN fine (other than searching of mail on the Vietnamese side)...