Start w/: use strict, use perl -w. Trust us, it'll really help you here. But, enabler that I am: you're mixing up your arrays and hashes. You're also trying to read FILE twice:
my @coufields = split(/\t/, <FILE>);
should eat all of FILE, leaving nothing left for the while loop. Leave that line out and try:

my $debug = 5;
print STDERR "Looking for id $courowid \n" if $debug > 8;
while (<FILE>) {
  chomp;
  my ($id, $course) = split(/\t/);
  print STDERR "trying id $id, course $course\n" if $debug > 8;
  if ( $id == $courowid ) {
    print STDERR "found id $id, course $course\n" if $debug > 5;
# if we match the ID, push the course name onto the 
# the array in the coudata hash, key $id
    push @{$coudata{$id}}, $course;
  }
}
print header;
if ( %coudata ) {
      # found, display data by taking each course back outof
      # the array
  foreach my $id ( keys %coudata ) {
    print STDERR "show id $id\n" if $debug > 6;
    foreach my $course ( @{$coudata{$id}} ) {
      print STDERR "show id $id, course $course\n" if $debug > 8;
      print ul("$id: $course");
    }     # foreach my course 
  }       # foreach my id
}else {
      # not found, show error
      print h1("Can't find row '$courowid'");
}
[download]

Its a big step in perl, but names are only half the battle: %coudata and $coudata{14146} are related but the first is the entire hash, the latter a single element found by key '14146'. @coudata is an entirely different kettle of data, its an array and the element $coudata[0] has no relation to the hash element $coudata{0}; just as @coudata doesn't have any relation to %coudata.
Plus, your variable names are troublesome: use english-ish things: course_list, course_id, search_id etc. You'll avoid typos (courdata coudata) and it'll make more sense. Just 'cause you can, doesn't mean its a good idea to to name an array and a hash the same thing. Its generally a bad idea, esp. early on. A few more keystrokes won't kill ya'

a