'I won't' is correct, but 'I don't' is not. For a promise for the future, we can only use 'will'/'won't'. You can never use 'do' for a future promise or offer. 'Do/don't' (present simple) is used for constant situations, repeated habits in the present, and for fixed arrangements for the future. Another alternative in British English, if you want to sound nice and old-fashioned, is 'I shan't' - the contraction of the first person 'I shall not'. Person A: That's a secret. Don't tell anyone about that. Person B: Don't worry. I shan't tell anyone.

They aren't both correct! I won't makes sense. I don't... doesn't. Don't is not used in this context!

The first doesn't make since. Don't is short do not, and won't is short for will not. The difference is don't means that you are not doing now, while won't means that you will not do it in the future. Don't can also mean that you are telling someone not to do something.

Person A: That's a secret. Do not tell anyone about that. Person B: Do not worry. I DON'T tell anyone about that.> not right. I do not want to tell anyone about it> its correct , but wrong for the context Or Person B: Do not worry. I WON'T tell anyone about that.> ok . Will not indicates the future tense > you are not going to tell anyone in the future.

In my opinion, they all seem true.