Isn't $printnumber the object reference??

So, you have to use ordinary operators for ordinary variables to make it do what you want. Here that would be something like

$printnumber += 1;
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$printnumber++;
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Just for fun, I'll make $printnumber an object, and have it accept your syntax:

#!/usr/bin/perl -w
{
    package NumberObject;
    use overload '""' => \&as_string, '0+' => \&as_number, '+' => \&ad
+d;

    sub new {
        my $class = shift;
        my $value = shift;
        bless \$value, $class;
    }

    sub as_string {
        return "" . ${+shift};
    }
    sub as_number {
        return 0 + ${+shift};
    }

    sub add {
        my $self = shift;
        return ref($self)->new($self->as_number + shift);
    }

    sub inc {
        my $self = shift;
        $$self += shift;
    }
}

$\ = "\n";
my $printnumber = new NumberObject(-1);
print "initial value:";
print $printnumber;
print "looping:";
for (1 .. 10) {
    inc $printnumber 1;
    print $printnumber;
}

print "overload magic (adding 5):";
$printnumber += 5;
print $printnumber;
print "still an object (adding 2):";
inc $printnumber 2;
print $printnumber;
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initial value:
-1
looping:
0
1
2
3
4
5
6
7
8
9
overload magic (adding 5):
14
still an object (adding 2):
16