use warnings;
use strict;

local $/ = "";
my $buffer;
while (<DATA>) {
	if ( s/\\section\{.*?\K(\\qr\{.*?\})(?=\})//i ) {
		$buffer .= $1;
	}
	elsif (defined $buffer) {
		s/^\S+\s+\S+\K/$buffer/;
		undef $buffer;
	}
	print;
}
print $buffer if defined $buffer;

__DATA__
\section{Results\qr{text ... text}}

Normal paragraph text here...

\section{Funding\qr{text ... text}}

Funding text here...