Sadly, Funnelback does not do anything like that out of the box. However, you can achieve something similar to what you mentioned by using filters to split the document and result collapsing to combine it back together.
Something like:
Create a filter which splits one html document into multiple documents where each document represents a section. As part of the splitting process, record some ID to determine which document it was split from.
At query time, enable result collapsing based on the ID of the source document. This ensure that only one document is shown for each source document.
There is a caveat though. Splitting up documents means you will lose anchor references which may negatively affect rankings. This approach would be suitable for things like indexing textbooks with multiple chapters but less idea for general websites.