Hi Üzeyir,
I'm an English teacher here on iTalki. It is quite unnatural, though in theory it does make grammatical sense. It doesn't really have any meaning that you're likely to want to say.

I think you would probably mean: I am too busy TO listen to your lies.
Lies are a bad thing; we know that they are not true, right? So we wouldn't think that listening to them had any value, therefore listening to them would be a waste of time. Are you with me? ('Are you with me?' is a way of saying 'Do you understand so far?' but it sounds much more polite.)
So someone could be too busy TO LISTEN to lies. But I can't see how somebody could be too busy to NOT listen to lies, because NOT listening to lies would be a good thing that would help you to be LESS busy.

I'll give you some examples of how you might use similar structures:
"I'm too busy to fix the car so I took it to the mechanic."
"I'm too busy to answer the phone and listen to my mother complain for two hours every day."

The only times that you could REALISTICALLY say "too busy to NOT (do something)" is when the last verb helps you to save time, e.g.:
"I am too busy to not hire a cleaner. I don't know how I'd get through the week without her."
"I am too busy to not get a good night's sleep, so I need to make sure I don't drink caffeine in the afternoons."

I hope that helps, let me know if you have any further questions. :-)

I would rephrase it as:

I'm too busy to hear your lies.