What impact has the Chinese Government had on Chinese Buddhism so far? State Administration for Religious Affairs (SARA) and the Buddhist Association of China are in charge of regulating religion in China.

So far, what effect have they had on doctrine for Chinese (non-vajrayana) Buddhism?

It's already well known that the Chinese government is hostile to the Dalai Lama.

(Also, as a side note, for those skeptical of any government's ability to regulate belief and practice- some percent of American Zen's doctrine and practice can be traced to Meiji era government initiated reforms)
I've seen quite a few news reports that focus on the huge growth of Christianity in China so naively I would have expected that this would now have overtaken Buddhism. However this report states that 


  The growth of Christianity is impressive, but Buddhist growth is
  extraordinary, especially with the country's history of official
  ideology of atheism


The figures are


18% identify as Buddhist
15% identify as Atheist
3.2% identify as Christian


Considering the vigorous supression during the cultural revolution (all monasteries closed, lands confiscated) it's notable that Buddhism has remained a substantial religion within China. However the attitudes of the Chinese authorities does appear to have softened and Buddhist are seen as a source of societal good. The Chinese government is clearly still very capable of suppressing religious movements (e.g. Falun Gong) and cultures (Tibetan) it finds threatening. Fortunately Buddhism doesn't appear to currently be a focus for this suppression.

As a reference note: This is a summary of the section on Buddhism in China from An Introduction to Buddhism by Peter Harvey (pp 409 - 410 in my copy).