The concern is not just over the money fueling wars but also the conditions in mines in Myanmar. A recent article in the Scotsman offers a chilling portrait of mining in the area. Allegations include the fact that mine owners may get workers addicted to drugs so that they can control them better.
Bans on Myanmar gems are nothing new. There has been a ban in place for several years, since the Congress banned the importation of products from the country. Tiffany & Co stopped buying rubies from Myanmar in 2003 and Cartier and Bulgari have joined them. Also the Jewelers of America has asked Congress to amend the ruling of the original ban to specifically mention gemstones mined in Myanmar. A Reuters article mentions that the reason for this is because some gem dealers are not taking the issue seriously. A quick scan of eBay revealed many listing for "Burmese rubies." Some of these are likely not from the region, unscrupulous dealers often tag rubies as "Burmese" because the rubies have such a reputation for beauty.
For a counterpoint, check out this post from Gary Roskin who writes for JCK and is a gemstone expert. He is in contact with dealers in the region who say that the gemstone money doesn't go to the government and that instead of hurting the government, a ban could hurt the Burmese people.
Despite the controversy, Myanmar has said that they are going forward with a planned auction of gems and jade next month. Unlike the regimented world of diamonds, the colored gemstone industry is far less centralized and regulated. There is also no Kimberley Process that addresses the human rights concerns for gemstones.