Since you're using a hash to hold the object, reusing the keys to map to bit positions might help. For example:

@fields = qw(organization report foo bar baz);
$bit = 1;
foreach $field ( @fields ) {
    $bitmask{$field} = $bit;
    $bit <<= 1;
}

This reduces the code that build $mask to

foreach $field ( @fields ) {
    $mask |= $bit{$field} if $obj{$field} !~ /^\s*$/;
}

The challenging part is converting the static regular expressions to something dynamic. Here's one (untested) thought:

sub testMask {
    my($bits, $on, $off) = @_;
    my($onmask, $testmask) = 0;

    foreach $field ( @$on ) {
        $onmask |= $bit{$field};
        $testmask |= $bit{$field};
    }
    foreach $ field ( @$off ) {
        $testmask |= $bit{$field};
    }
    return ($bits & $testmask) == $onmask;
}

This lets you rewrite the tests as:

moreDatabaseProcessing() unless testMask($mask, \qw(organization), \qw(foo bar));

The up side of this approach is that it's completely position independent. The downside is that it's not resilient against field name typos, though that can be mitigated with some extra tests.