Problems:

• /[4]/ matches "4", not "[4]".

• You never exit the inner loop before the end of the file.

• $errorCode != 0 || $errorCode != 1
  should be
  $errorCode != 0 && $errorCode != 1
  The former is always true.

• die ... unless -f "$logFile"; is useless, even harmful. open not only performs this check, but can provide a better error message.

• You call <> (in the outer loop) after being told you've reached the end of the file (in the inner loop).

• \[ is more usual and readable than [[]
  \] is more usual and readable than []]

• while (...) { ...; next }
  is the same as just
  while (...) { ...; }

Fix:

use strict;
use warnings;

my $ArcotIDError_Count = 0;

OUTER:
while(my $line = <>) {
    $line =~ tr/\r\n//d;
    next if $line !~ /Handling NSPAdvice for mechanism \[4\]/;

    do {
        $line = <>;
        last OUTER if !defined($line);
    } while $line !~ /Authentication mechanism returned \[(\d+)\]/;

    my $errorCode = $1;
    #print("\$errorCode=$errorCode\n");
    if ($errorCode != 0 && $errorCode != 1) {
        $ArcotIDError_Count++;
    }
}

print "Total Number Of ArcotID Authentication ErrorCode returned is $A
+rcotIDError_Count\n";
[download]

( I see that others have identified some of the issues, but not all. )

Update: Reordered points by descending severity.