use strict;
use warnings;

my $data = do{local $/; <DATA>};

my $i = 1;
while ($data =~ s/(<FN$i>(?:(?:(?!<\/FN$i>).)*)<\/FN$i>)\n*//gsi){
	my $foot = $1;
	$data=~ s/<FN${i}ref>/$foot/gi;
	$i++;
}
print $data;

__DATA__
This is my text<FN1ref>. This is my text
This is my text. This is my text<FN2ref>. This is my text
This is my text. This is my text<FN3ref>, this is my text
This is my text. This is my text<FN4ref>, this is my text
This is my text. This is my text<FN5ref>, this is my text
This is my text.
<FN1>Footnote1</FN1>
<FN2>Footnote2</FN2>
<FN3>Footnote3</FN3>
<FN4>Footnote4</FN4>
<FN5>Footnote5</FN5>

Output:
-------
This is my text<FN1>Footnote1</FN1>. This is my text
This is my text. This is my text<FN2>Footnote2</FN2>. This is my text
This is my text. This is my text<FN3>Footnote3</FN3>, this is my text
This is my text. This is my text<FN4>Footnote4</FN4>, this is my text
This is my text. This is my text<FN5>Footnote5</FN5>, this is my text
This is my text.