The WiFiWebServer tutorial describes how to use the ArduinoWiFiShield. This shield is based on different WiFi hardware than the ESP8266 modules.
You need a WiFi library driving the ESP8266 modules instead, e.g. the one mentioned in http://contractorwolf.com/esp8266-wifi-arduino-micro.
Your favourite internet finding machine sure will find lots of similar libraries.
A different approach would be ArduinESP, which means to develop code to run on the ESP8266 module directly.
...aaaand please rethink your wiring. The ESP8266 modules are a pure 3.3V breed and your wiring connects a 5V logic signal directly to the ESP8266 and this may send your ESP8266 into the digital heaven with or without waving goodby using blue smoke...