Get rid of all the square brackets from your regex because you do not have any in your input string:

use warnings;
use strict;

my $sentence = "Furthermore , expression of <GENE> Vpu </GENE> in Jurk
+at T cells rendered them more susceptible to <GENE> Fas </GENE> - ind
+uced death ";
$sentence =~ s/<GENE>\s*(((?!<GENE>|<\/GENE>).)+)\s*<\/GENE>//gi;
print "$sentence\n";

__END__

Furthermore , expression of  in Jurkat T cells rendered them more susc
+eptible to  - induced death
[download]

Running your regex through YAPE::Regex::Explain highlighted the square brackets