"Do not track" is actually an HTTP header (DNT). So you should be able to do this by adding that header to every request by way of loadUrl(url, headers). For example:
webView.setWebViewClient(new WebViewClient()
{
@Override
public boolean shouldOverrideUrlLoading(WebView view, String url)
{
HashMap<String, String> extraHeaders = new HashMap<String, String>();
extraHeaders.put("DNT", "1");
view.loadUrl(url, extraHeaders);
return true;
}
});
Note that this loadUrl() overload requires API level 8.