For pipelining to work, responses must come back in the order they were requested. A naive server implementation might just send the response as soon as it has been calculated. If multiple requests are sent in parallel, and the first request one takes longer to process (e.g. processing a larger image), then the responses will be out of order.
This is a problem for the client since HTTP is a stateless protocol, the client has no way to match the requests with the responses. It is reliant on the order the responses came back in.
Even if the server does properly support pipelining, performance issues can arise because all subsequent requests have to wait for the first one to be complete (Head of Line blocking).
This article, talks about performance loss in some circumstances and a potential of denial of service attack.
This article also suggest that pipelining isn't a massive win.
WWDC 2015 - Networking with NSURLSession explains head of line blocking really well. (The solution is to switch to HTTP 2 which support priorities)
So in summary the issues with HTTP pipelining are: