I went to Beijing in 2009 and found it to be sonically very rich. The first thing that struck me was the number of people - sounds like a cliché but the population difference between Beijing and the big cities in Europe is quite apparent. This makes recording nature (such as in parks, of which there are many) slightly difficult. On the flip side, markets, subway stations, teahouses, streets etc. make very interesting subjects to record.
As Edmatthews82 has said, just be aware of recording in certain places. I did all recording with a small handheld. Check this post for more info. There are many street markets in the daytime and at night which are pretty cool to record. Silk St market is a great place to get cheap counterfit clothes, and offers an interesting soundscape and Donghuamen, a night food market is also well worth a visit. Also, the Hutongs (complex network of small alleys and backstreets) are usually pretty lively and I got some great recordings there.
In terms of temples and monuments, at the Forbidden City, there are usually some musical performances and a lot of interesting temple sounds (gongs, chanting, traditional instruments). Further towards the outskirts, the Summer Palace is a much bigger, greener space where it's possible to record much calmer ambiances. But there are many smaller temples, such as Prince Gong's palace, where you can often find some interesting (and usually very calm) soundscapes.
Teahouses are great places to record, as well as to soak up some local atmosphere. Try to find a genuine one though, rather than a tourist-aimed operation. I found the soundscapes of the subway system to be really interesting too. Maybe it's the fact that to me, the language is so different, but the announcements reverberating around the station, along with the hustle and bustle of people made it very rich.
Lastly, if you have enough time it's well worth making a trip to the Great Wall of China. There are a couple of spots not far from Beijing, but these tend to be very busy. I would recomend taking a trip a bit further out to avoid the masses. I went to Jinshanling (about 2.5 hrs from Beijing) and hiked 12km to Simatai. An awesome experience both sonically and visually.
I have archived all of my recordings on my Sound Map. If you're interested to hear what I recorded, just zoom in on Beijing and have a listen. Enjoy your trip, and I'd love to hear what you experienced once you're back!