# Always use strict especially when you are new. Its a boon to debugging and it helps you from screwing up in some interesting ways (but mostly it does some typo-checking)
use strict;
use warnings;

open (TEXTFILE, 'text.txt') or die ("Cannot open file : $!");

# $/ is normally "\n" which is what readline() uses to decide when to stop reading. You want to assign undef here so that readline() will pull in the entire file and not just the first line of the file.
$/ = undef;

# strict will force you to declare your variables. Here I use my() to make $big_string be a lexical.
my $big_string = <TEXTFILE>;
 
$big_string = lc ($big_string);

# I re-wrote this as a tr/// expression so it would be so ugly as a regular expression. If you really wanted to use s/// then s/[()]//g would have been preferred.
$big_string =~ tr/()//d;

# I declared @words with my() and moved the plus sign out of the character class. You meant to split on repeated white-space but you actually just indicated that plus signs should also be split on. Moving the + quantifier onto the character class indicates that any repeated matching of the class should be smushed together.
my @words = split (/[,.\s]+/, $big_string);

# I replaced your for/push loop with a grep which only passes words that are three characters or longer. You should also note that in your original expression that anything that matches {4,} is also going to match {3,}. If you really meant to do that then just the single {3,} would have been good enough. Also, instead of saying [a-zA-Z] you could have written /[a-z]{3,}/i which is case insensitive. 
my @gt_three_char_words = grep /[a-z]{3,}/i, @words;

# Rename %hash to %word_counter and declare it.
my %word_counter;
foreach (@gt_three_char_words) {
    $word_counter{$_}++;
}

# Sort the list of words so you can then extract whatever portion you like.
@gt_three_char_words = sort {
        # Sort the word list by occurrances
        ( $word_counter{$a} <=> $word_counter{$b} ) ||
        # Then alphabetically
        ( lc( $a ) cmp lc( $b )
    } @gt_three_char_words

# Print the 
print
    join( '',
          map( "$_: $word_count{$_}\n",
               @gt_three_char_words[ 0 .. 9 ] ) )
    . "\n";

##</code><code>##

use strict;
use warnings;
use Getopt::Long;
use File::Slurp;

my $filename = 'text.txt';
my $min_word_length = 3;
my $start_word = 0;
my $words_to_display = 10;
GetOptions(
    "file=s" => \$filename,
    "min=i" => \$min_word_length,
    "start=i" => \$start_word,
    "words=i" => \$words_to_display
);

my $original_text = lc read_file( $filename );
$original_text =~ tr/()//d;

my @words =
    grep /[a-z]{$min_word_length,}/i,
    split /[,.\s]+/,
    $original_text;

my %word_counter;
$word_counter{ $_ } ++ for @words;

@words = sort {
        $word_counter{ $a } <=> $word_counter{ $b } ||
        $a cmp $b
    } @ words;

my @words_to_print =
    @words[ $start_word ..
            ( $start_word + $words_to_display - 1 ) ];

for my $word ( @words_to_print ) {
    print "$word: $word_counter{ $word }\n";
}