Is the #file a reference to my data file?

No, not directly. Instead, the idea presented was to capture the whole file in @file:

my @file = <DATA>;
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In your case, it would be <INFILE> instead of <DATA>. (<DATA> is a special file handle, starting after the special tag __DATA__ in the script, but otherwise pretty much a normal-behaving file handle.)

The $# in front of an array identifier means "the last index of" that array.

Perhaps better is to keep the last three lines in a fifo:

open(INFILE, "< file.log")
         or die "Cant open file : $!";

open(OUT, "> results.txt")
         or die "Cant open new file : $!";

$pattern = "Error code 2";

my @file;
while (<INFILE>) {
    print OUT @file if (/$pattern/ and @file);
    push @file, $_;           # push on the current line
    shift @file while (@file > 3); # shorten to 3 elements
}

close INFILE;
close OUT;
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Don't forget warnings and strict.