When do we use each of the following?
I don't want to ...
I won't want to ...
I wouldn't want to ...
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Don't means do not, won't means will not, and wouldn't means would not.
This per se means you currently don't want to do it; however, for a native speaker, if X is defined to be a future event, it can mean "would not" (e.g. I don't want to do X even if I get the chance). However, for a native speaker, a more natural-sounding way to say this would be "I don't think I'll want to do X."
This means that as far as you now know, when in future the chance comes to you, you will not want to do it. It implies that the chance will at least probably come.
This means if in future the chance comes to you, you will not want to do it. It implies that the chance might not ever come. Would is usually qualified with an "if" clause, e.g:
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