Unfortunately, Android does not support viewing PDFs out of the box in a WebView. Luckily, Google has a nifty little tool that allows you to perform this very task quite easily using Google Docs. Basically we will embed our PDF in a Google Doc page on-the-fly and load that. Here's the code:

String myPdfUrl = "http://example.com/awesome.pdf";
String url = "http://docs.google.com/gview?embedded=true&url=" + myPdfUrl;
Log.i(TAG, "Opening PDF: " + url);
webView.getSettings().setJavaScriptEnabled(true); 
webView.loadUrl(url);

Simply replace the myPdfUrl value with your own remote PDF, and you're done.