To me, the above translates as...

I want to ask your lips, I want to ask your language.

But it doesn´t make any sense.

Help!!!

Dil also means tongue.

Don´t forget the nose

you talk to your nose ?

I respect nose a lot. It is in a higher position than lips and tongue. First I would ask the nose

1)Dudaklarını sormak istiyorum

In this sentence dudaklarını is a definite object(accusative). Sentence means I want to ask about your lips.

2)Dudaklarına sormak istiyorum

In this sentence dudaklarına is indirecet object(dative). Sentence means I want to ask (to) your lips. You don´t use "to" here. But I put it there to explain better.

In English if you say "I want to ask your lips." I guess it can mean both of them. Right?

Very amusing guys...but please don´t confuse me anymore than is absolutely necessary!!

Is it normal, in Turkish, to want to ask the lips and tongue?

Don´t you just say..."I want to ask YOU something" ?????

He doesn´t want to talk to them. He asks about them. English makes it confusing.

Why would you add "senin" at the beginning of this sentence?

Is it merely for extra emphasis?

Yes. There is no need for it because there is a possesive suffix at the end of dudaklar-ın-ı. This possesive suffix can only adress "you" So there is no need to add "senin".