“I don't know about that.”
Your friend invited you to go out to a dance club. You don't really like to dance, and you don't really want to go, but you're not 100% against it. You say this to be polite.
I don't know about that.
You use this phrase when you don't agree with something that someone says. But this phrase also doesn't mean that you're completely against what they're saying. So it's good for situations where you disagree, but you're a little unsure.
This can be used when someone wants you to do something that you don't want to do, like in the example above. Or it can be used when someone makes a statement that you don't think is correct. For example:
A: I think you're supposed to turn left at the next light.
B: Hmm... I don't know about that.