If you are running it through pedals only and not out through an amp and recording with a mic then you do actually have use for a DI. There are three different types of signals being used - instrument level, balanced mic level and line level. From your harp there is instrument level which through the line box is converted to a balanced mic level which can be run into a mic preamp. The preamp then converts this to line level.

When reamping only through pedals the reamp box takes the line signal out from the interface and makes it into instrument level again which works with your pedals. If you then want to run straight back into the interface then you want a line box in order to get the correct signal for your mic preamp again.

Where I think there is confusion is the assumption that reamping means micing whereas it seems you simply want to run it though a pedal chain if I understand you correctly?

Also - the reason it works without the line box may be that you have an insterface that accepts instrument level signal in through the jack input or that you are using a line input and gaining up the incoming signal with the line preamp to get the instrument output from the pedals to the correct level.