Will it be a problem if I enter with an unexpired visa but leave on an expired visa?

I guess you mix the validity of China visa and duration of each stay after entry, please refer to https://www.travelchinaguide.com/images/photogallery/2009/chinese-visa.jpg to read your visa. For example, your visa is issued on March 1, 2012, the enter before date is June 1, 2012, duration of each stay 030 days after entry on your visa, you enter China on May 30, 2012, you still can stay in China for 30 days counted from May 30, 2012. It is not an expired visa after June 1, 2012 if you enter China before June 1, 2012. June 1, 2012 is the last date for you to enter China, but the duration of stay is counted from the date of entry.