"The" ne "the". You should normalise the case.

Other issues:

• You use use warnings;. Great! You should also use use strict;, though.

• Why loop when you know you're only going to get one value?

  undef($/);
  while (<FILE>) {
  }
  [download]

  should be

  undef($/);
  $_ = <FILE>;
  [download]

• Use of alternation inside character class.

  s/[\,|\.|\!|\?|\:|\;|\"|\'|\<|\>]//g;
  [download]

  is the same as

  s/[\,\.\!\?\:\;\"\'\<\>||||||||]//g;
  [download]

  and

  s/[\,\.\!\?\:\;\"\'\<\>|]//g;
  [download]

  You want an alternation

  s/\,|\.|\!|\?|\:|\;|\"|\'|\<|\>//g;
  [download]

  or a character class

  s/[\,\.\!\?\:\;\"\'\<\>]//g;
  [download]

• Useless escaping. For readability,

  @array = split(/\ /, $_);
  
  s/[\,\.\!\?\:\;\"\'\<\>]//g;
  [download]

  should be

  @array = split(/ /, $_);
  
  s/[,.!?:;"'<>]//g;
  [download]

• '<insertfilepath>' || $!
  [download]

  is the same as just

  '<insertfilepath>'
  [download]

  since the file name will always be a true value.

• Useless use of global variables (FILE), and unlocalised changes to global variables ($/).

• Splitting on spaces won't split on newlines, and will produce empty strings when there are two spaces in a row. Split on special ' ' instead.

  @array = split(/ /, $_);
  [download]

  should be

  @array = split(' ', $_);
  [download]

use strict;
use warnings;

my $file;
{
   my $qfn = '<insertfilepath>';
   open(my $FILE, '<', $qfn)
      or die("Can't open \"$qfn\": $!\n");
   local $/;
   $file = <$FILE>;
}

my %word_counts;
for (split(' ', $file)) {
   s/[,.!?:;"'<>]//g; 
   ++$word_counts{lc($_)};
}

for my $word (sort keys(%word_counts)) {
   print "$word occurred $word_counts{$word} times\n";
}
[download]

Update: There's no reason to load the entire file into memory at once, and if you don't, you gain the ability to pass a file name on the command line.

use strict;
use warnings;

my %word_counts;
while (<>) {
   for (split(' ', $_)) {
      s/[,.!?:;"'<>]//g; 
      ++$word_counts{lc($_)};
   }
}

for my $word (sort keys(%word_counts)) {
   print "$word occurred $word_counts{$word} times\n";
}
[download]