Since you're already reading the entire input at once, read it all into a scalar with something like
    my $input = do { local $/;  <FILE>; };
and then

c:\@Work\Perl\monks>perl -wMstrict -le
"my $input = qq{line1\nline2\n\nline4\n\n\nline7\nline8\n};
 print qq{[[$input]]};
 ;;
 $input =~ s{ ^ \s* \n }{}xmsg;
 print qq{<<$input>>};
"
[[line1
line2

line4


line7
line8
]]
<<line1
line2
line4
line7
line8
>>
[download]

Update 1: Or, if you need to read all the lines of the file as separate lines and keep them that way, try
    my @lines = grep !m{ \A \s* \Z }xms, <FILE>;

c:\@Work\Perl\monks>perl -wMstrict -le
"my @lines = (
   qq{line1\n}, qq{line2\n}, qq{\n}, qq{line4\n}, qq{ \t\n},
   qq{\t \t \n}, qq{line7\n}, qq{line8\n},
   );
 print qq{[[@lines]]};
 ;;
 @lines = grep !m{ \A \s* \Z }xms, @lines;
 print qq{<<@lines>>};
"
[[line1
 line2

 line4


 line7
 line8
]]
<<line1
 line2
 line4
 line7
 line8
>>
[download]

Update 2: Slight formatting fix to code in Update 1; no functional change.

Update 3: Rather than using
    !m{ \A \s* \Z }xms
to grep lines in the code in Update 1, I think I'd prefer
    m{ \S }xms
as used by Marshall here; don't know why I didn't use it in the first place. Note that no logical negation is needed.