How to use "chase" meaning to remind to do something
Is it usual or appropriate to use in business English "chase" meaning to remind to do something?
Does "chase up" have the same meaning?
Could you please give some example how to use it so that it sounds naturally

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It's common in the UK but I can't speak for the USA.
e.g. I'll chase up that late invoice.  Can you chase them (up) for an answer?

You must use "chase up" with a thing but you can choose either for a person.

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In the US we don't use "chase" or "chase up" in this context.

If you are trying to find something, you might use "chase it down" in certain contexts.

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Chase is not used in this way in English, "chase up" is also not used in English.  Hope that helps!