Aylardır burnumda tütüyorum.

 This sentence is not right..I mean that should be "  Aylardır burnumda tütüyor" if it is so then  " I am missing/yearning for it/she/he so much.

burnunda tütmek literarly means to give out smoke in the noise, but its a saying, this saying means to yearn for, to miss someone or something alot, to crave for

Aylardır burnumda tütüyor = I am missing it/her/him so much for months.

Aylardır burnunda tütüyorum = I am being missed by it/her/him so much for months.

Aylardır burnunda tütüyor = she/he is missing it/he/she so much for months

Aylardır burunlarında tütüyor = They are missing it/she/he so much for months

Aylardır burnumuzda tütüyor = We are missing it/she/he so much for months

And I want to add something: it doesn´t have to be "aylardır" only. it can be other time phrases too. Yıllardır (for years), günlerdir (for days), mevsimlerdir (for seasons) can be other examples.

By the way, this is mostly used by the first singular or plural person (ben/biz), or for the third person (o/onlar) -for stories-

For that reason, "burnumda tütüyorsun" sounds more acceptable as its meaning; "I am missing you" 

"burnumda tütüyorum" means im missing myself. "burnumda tütüyorsun" means im missing you. its an expression/proverb. direct translation would be something like "youre burning on my nose" which is highly meta

Wouldn´t it be "Burnumda yanıyorsun" if we translated it back?