First, let's make your code readable by running it through perltidy:

#!/usr/bin/perl
#
use strict;
use warnings;

use List::Util q{first};

sub search_phrase {
    my @array;
    my ($inFile, @phrases) = @_;
    my $lastPhrase = $phrases[-1];

    open my $inFH, q{<}, $inFile or die qq{open: $inFile: $!\n};
    my @lines = <$inFH>;
    close $inFH or die qq{close: $!\n};
    foreach my $phrase (@phrases) {
        my $rxPhrase = qr{\Q$phrase\E};
        my $lineNo   = first {$lines[$_] =~ $rxPhrase} 0 .. $#lines;

        unless (defined $lineNo) {

            #print "-1";
            #        print qq{$phrase: not found in sequence\n};
            next;
        }
        print "" if ($lines[$lineNo] =~ m{\Q$lastPhrase\E\s*(\d*)});

        push (@array, $1);
        $lineNo++;
        splice @lines, 0, $lineNo;

    }
    return (@array);

}


my $file_n  = "665172.data";
my $phrase1 = "total rejected rows:";
my $phrase2 = "total rejected recors:";
my $phrase3 = "rejectb:";
my $phrase4 = "total rejected rows:";

my @newarray =
      search_phrase(
                     $file_n, $phrase1, $phrase2, $phrase3,
                     $phrase4
                     );
my $count = ($#newarray);
$newarray[$count] =~ s/\s+//g;

if ((($#newarray + 1) >= 1) && ($newarray[$count] gt 0)) {
    print "$newarray[$count]\n";
} else {
    print "-1\n";
}
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Next, when I try running your code, it works fine, and you say it works fine for you, except when you ru it on a server.

So the obvious question (but one that I'll ask anyway) is, "What's the difference from your workstation and the server?" You're only using List::Util -- I imagine it's installed on the server, and the same version number. Is the data file the same?

Now onto the fun part: picking out stylistic miscues.

• You're reading the entire file into an array. it's more Perl-ish to loop around the diamond operator: (updated: please see footnote.)

    foreach my $thisLine (<$inFH>) {
    .. do something with $thisLine ..
    }
  [download]

    while (<$inFH>) {
    .. do something with $thisLine ..
    }
  [download]

• You're using a variable for the filehandle -- this isn't really necessary. I suggest using something like INPUT instead.
• You don't have the main-line enclosed in a set of braces. I like to do that, if only to remind the reader that it's the place where the action is taking place -- as opposed to some location where we're defining variables.
• You're not describing what you're doing. Without spending a lot of time analyzing your code, it seems you're looking for key phrases in a file and grabbing the numerical value.
• Whenever I see $phrase1, $phrase2, $phrase3 and so on, I immediately wonder, "Why isn't this program using an array?"

Those are my thoughts. What do you think?

Update: As both Narveson and wfsp have pointed out, for reads the entire file in, so doesn't achieve the goal I was reaching. Instead, use while. Oops.