Here is a different approach which gives you the output you want:

use warnings;
use strict;

my $i;
my $scaf;
while (<DATA>) {
    if (/^>(.*)$/) {
        $i = 0;
        $scaf = $1;
        print ">$scaf.$i\n";
    }
    elsif (/^N/) {
        $i++;
        print ">$scaf.$i\n";
    }
    else {
        print;
    }
}

__DATA__
>Long_DNA_Scaffold
atgctagsctgtagctagctagtcgatgctagtagct
ttagctagctgatgctagtcgatgctagtcatctagc
NNNNNNNNNNNNNNNNNNN
tagctagctagctgatcgtagctagtcgtagctagct
ggtcgatgctgatcgtagctgatgctagtcgatgcta
>Long_DNA_Scaffold2
atgctagsctgtagctagctagtcgatgctagtagct
tagctagctgatcgtagctagctgatgctagctagtc
NNNNNNNNNNNNNNNNNN
tagctagctagctgatgctagctagtcgatcgtagct
ggctagtgctagtgctagctgatgctagtcgtagctg
[download]

Prints:

>Long_DNA_Scaffold.0
atgctagsctgtagctagctagtcgatgctagtagct
ttagctagctgatgctagtcgatgctagtcatctagc
>Long_DNA_Scaffold.1
tagctagctagctgatcgtagctagtcgtagctagct
ggtcgatgctgatcgtagctgatgctagtcgatgcta
>Long_DNA_Scaffold2.0
atgctagsctgtagctagctagtcgatgctagtagct
tagctagctgatcgtagctagctgatgctagctagtc
>Long_DNA_Scaffold2.1
tagctagctagctgatgctagctagtcgatcgtagct
ggctagtgctagtgctagctgatgctagtcgtagctg
[download]

Other comments:

• It is good that you used warnings. You could also try to use diagnostics to get a better explanation of the warning messages.
• Read the doc for tr///. It is not clear what you are trying to do.
• It is also good practice to use strict.
• See also Basic debugging checklist.
• Read How do I compose an effective node title?. If you edit your node title to include some keywords like DNA, you might get a better response from the Bio monks.