welcome in H.C.M.
I have gone through your query regarding your cough
with green sputum with taste like burning rubber.You have already consulted GP but no satisfactory result came. No doubt you are having respiratory tract
infection and this infection also involving the posterior part of tongue where taste buds
are also involve to give you smell of burning rubber.
Considering your problem you are advice for culture and sensitivity test of your sputum to find out infective organism and the name of antibiotic against your infection. When this correct antibiotic will be given to you with some mucolytic agent
you will be benifitted.
Only this much help is possible by analysing the complaints if you feel not much relief by above treatment than physical examination by specialist becomes necessary to find out further line of treatment.
Hope your query is now clarified.