It is not what is executed in the else block, but what is execute after.

if (expression) {
    statement (1);
    }
else {
    statement (2);
    }
statement (3);
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If statement (1) exits the enclosing scope with return/exit/die/croak, statement (3) is never executed. An if/else is the to do something different based on a condition and then continue doing what is after the construct for all cases. This implies that an else immediately after return/exit/die/croak is always useless.

The original code does it right in the first block:

sub get_filehandle {
    my $self = shift;
    $self->{file} = shift;
    open my $fh, "<", $self->{file}
        or die "can't open $self->{file}";
    return $fh;
    }
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Which - when following the wrong method you approve of - would be

sub get_filehandle {
    my $self = shift;
    $self->{file} = shift;
    if (open my $fh, "<", $self->{file}) {
        return $fh;
        }
    else {
        die "can't open $self->{file}";
        }
    }
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In my way, the second block should be reduced to

sub get_lines {
    my $self = shift;
    $self->{file} = shift;
    ### get the filehandle if we don't already have one
    $self->{file_handle} ||= $self->get_filehandle ($self->{file});
    return readline ($self->{file_handle}) || undef;
    }
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That last return is arguable anyway, as the readline method can be a method that legally can return undef, "", and 0 (or any object evaluating as false) indicating that the original code was wrong to start with.