Claudio, The real problem is that you are using “would” differently in the two sentences. In 1) You are using would to express possibility or presumption about an event in the past. I didn't know I would have been asked to pay the taxes when checking-in. I don’t know that I would be asked to pay the taxes when checking-in…..present tense version. As Scott stated it is a mixed conditional. Mixed conditionals can have very complex meanings. In the sentence you were only asked to pay the taxes theoretically. You could restate it like this: I didn't know that if I had been checking-in, I would have been asked to pay the taxes. or I didn't know that If I had checked-in, I would have been asked to pay the taxes then. In 2) You are using would to express futurity from a point of view in the past. I didn't know I would be asked to pay the taxes when checking-in. I don’t know that I will be asked to pay the taxes when checking-in…..present tense version. In this case you are simply saying that you didn’t know beforehand that you would be asked to pay the taxes when checking-in. When used in this way “would” maintains its connection with the verb “will”. Historically would is the past tense of the verb will. You can see it more clearly when changing statements to indirect speech. “ I don’t know if I will be asked” – he said. He said that he didn’t know if he would be asked.

Yep, there are differences. #1 is INCORRECT. #2 is CORRECT. That's the difference.

The first sentence is mixing tenses in my opinion, the second sentence is correct.