Yes many. Here is a list.
1. You do not have control of your applications and data.
2. When a server goes down, everything goes down. You cannot do anything about it. Recently, GoDaddy Server went down and thousands of websites were unavailable for hours. Also, Amazon cloud service went down recently and many websites did not work.
Basically, you are relying on somebody else. but the good thing is, it is cheaper and you don't need to manage anything.