I applied to a PhD program a few weeks after the university's deadline for this year's scholarship, TAship, and RAship application passed. The application for a scholarship from external funding bodies also closed before I finalized my PhD application. Today, I received an invitation for an interview next week regarding my PhD application. The problem is, even if I get admitted into this PhD program, I would not be able pursue it for lack of scholarship. Should I respectfully decline the invitation, then?
Yes, you should decline to do an interview if there is any definite reason that you would not accept the position, to save time both for yourself and the interviewer(s).
However, this circumstance sounds a bit odd to me, and I'd recommend clarifying before/while declining the invitation. Something like:
My understanding is that deadlines for funding opportunities have passed. If it is true that my position cannot be funded even if I am accepted, I will need to decline the invitation. Please let me know if my understanding is incorrect.
Like Jon Custer mentions in a comment, there may be ways around these deadlines if the university wants to admit you, and if the lack of funding is the only reason you would definitely decline the position then it would be a shame to miss the interview on those grounds if in fact funding could be made available somehow.