To all those who have been able to solve the problem, you were lucky. Thank you for your response.
1- But my case is different. Once microphone of the headset is burnt out, it will stop working on every device including, but not limited to, MacBook, Windows laptops, iphone5, iPad4, Samsung Galaxy, not just a particular iphone6 Plus only. In short, the mic does burn out randomly when headphones are used with my iphone6 Plus. The headphone speakers continue to work normal.
2- Secondly, iPhone6, iPhone6Plus and iPhone6S are concealed devices with absolutely no user serviceable parts. Even battery is NOT swappable by end-user. Chances of further damaging the headphone jack are much higher when using screws, toothpicks, etc. to take out lints or for cleaning the Jack terminals. Probably, a purpose built blower/vacuum cleaner for iPhones can be helpful to suck-out any lint. Until that becomes available, I am not risking loosing my factory warranty. Fortunately, I don't see any lint in the headphone jack hole, unless it was inserted by Apple during the manufacturing that is not viewable by the end user.
3- I paid for the iPhone6+ directly to Verizon using my personal credit card and my employer never pays for iPhones nor reimburses them nor I was reimbursed for it. In short, I own it. Verizon took my personal home address for delivery address for the replacement set under warranty. However, Apple chose to give me yet another surprise and sent the replacement to my employer's headquarters! Hence, over the holidays, I had to continue to live with this misery. My employer has redirected the shipment to me and shall reach me soon.
4- I checked with my extended circles of friends. None of them is happy with recent models starting iphone6. The Jack problem seems to be inherent to all iphone6, iPhone6S and iPhone6 Plus - Hence it is confirmed to be a manufacturing defect, IMO.