Can a document, for which payment was not made at the time, and dealing with a friend who started his project with an interest-based loan, make the work unlawful?

The prohibition of an interest-bearing loan relates to the one who takes it, not to the specific item for which it is spent. Therefore, there is no harm in dealing with someone who purchased something with loan money, whether for compensation or without compensation. Their gift should be accepted, and their transactions of buying, selling, and renting are valid. The Prophet, peace and blessings be upon him, dealt with the Jews despite their consumption of usury. Consequently, there is no harm in the document your friend prepared for his project, which was financed by an interest-bearing loan, whether payment was made for it or it was a donation.