eval BLOCK catches die. The following catches warnings and other uses of STDERR:

use v5.8.0;
my $stderr;
{
   open(local *STDERR, '>', \$stderr);
   warn("foo\n");                    # no output
}
print STDERR ("\$stderr: $stderr");  # $stderr: foo
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Together:

use v5.8.0;
my $stderr;
eval {
   open(local *STDERR, '>', \$stderr);
   warn("foo\n");                    # no output
   die("bar\n");                     # no output
};
print STDERR ("\$stderr: $stderr");  # $stderr: foo
die($@) if $@;                       # bar
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In earlier version of Perl, I think you can use IO::Scalar to do:
tie *STDERR, 'IO::Scalar', \$stderr;