To be on a business trip. You do need the "a" here. "On" is also used for an excursion, a tour, a journey, and so on. However, you can be "in" a business meeting.

"to be on a business trip" But you may say "to be in a business trip situation like..." as "in a situation" is a phrase with "business trip" added in the middle. So it depends on what you want to say after, if anything. And pay attention that you need "a" [singular], unless you say "business trips" [plural] instead.

To be ON business trip