This method for playing FM on a mobile phone without headphones works only on Motorola W220.Here's what the phone looks like-

Just type *#**372# and press the call button.You will reach the FQC menu with options like S/W ver. &model,Battery info,Indication,FM Radio etc.Just select FM radio.The FM radio will play without asking you to insert the earpiece.The reception will obviously not be too good.Try placing your fingers at different parts of the phone body to get better reception.My mom has a W220 so I've tested it and it works.

For other phones,I would suggest you to try to get the code for the FQC menu to listen to FM without inserting the headphones.I would like to thank a friend of mine,Aditya Kusray for telling me the code for W220.