China has interests and involvements in Burma, but limited leverage. Burma is not some kind of client state of China. It is a xenophobic, divided, tribalized country with a nationalistic government; it bears more resemblance to one of the less coherent sub-Saharan African states than to most other East Asian countries. It’s not an easy place to influence. Through most of the 1980s there was a Burmese Communist Party, which consisted primarily of the Wa tribe plus Chinese leadership. When the Wa decided to turn anti-communist in the late 1980s and chased the Chinese leadership into China, China’s influence in the country was drastically reduced but there was little China could do without military intervention. So Beijing basically sat by passively when it happened.
So there, I guess.