I added a few comments to your script. I don't exactly know what outcome you want, but I do spot a few things that are most likely errors.

#!/usr/bin/perl -w

use strict;
use locale;
use warnings;
use diagnostics;
use utf8;

binmode(STDIN,  ":encoding(utf8)");  # prepend the colon to "encoding"
+.
binmode(STDOUT, ":encoding(utf8)");  # prepend the colon to "encoding"
+.
binmode(STDERR, ":encoding(utf8)");  # prepend the colon to "encoding"
+.

my $data_file = "data.txt";
my ($start, $end);

# Warning: this works, but DATA is special.  Better to use a different
+ name.
open (DATA, '<:utf8', $data_file) || die "Couldn't open $data_file : $
+!\n";

# Consider using IO::Prompt::Tiny, or IO::Prompt::Hooked.
print "Start time (format : yyyy.mm.dd.hh):\n";
chomp (my $start_input = <STDIN>);

print "End time (format : yyyy.mm.dd.hh):\n";
chomp (my $end_input = <STDIN>);

# Questionable: Are you really intending to pass any input that contai
+ns four
# contiguous digits?  Maybe so, but seems like an inadequate validatio
+n.
if ($start_input =~ /\d\d\d\d/ && $end_input =~ /\d\d\d\d/){
  while (<DATA>){
    # Even more questionable: None of your suggested input formats wou
+ld match.
    # Did you mean $start_input =~ /\d{6}/ ?  When would the input eve
+r have
    # six contiguous digits?
    if (/$start_input\d\d\d\d\d\d/ .. /$end_input\d\d\d\d\d\d/){
      if ($_ =~ /(.*)\t(.*)\t(.*)/){
        print "Date  = $1" . "\t" . "Temp_moy_DegCs = $2" . "\t" . "HR
+_moy_s = $3\n";
      }
    }
  }
}

if ($start_input =~ /\d\d\d\d\.\d\d/ && $end_input =~ /\d\d\d\d\.\d\d/
+){
  # Maybe you intended to do this immediately after receiving input.  
+That would
  # fix the second "if", but would break the fourth "if" conditional.
  $start_input =~ tr/\.//d;
  print "$start_input\n";
  $end_input =~ tr/\.//d;
  print "$end_input\n";
  while (<DATA>){
    # I don't see this ever working.  Did you mean $start_input =~ /\d
+{4}/ .. ?
    if (/$start_input\d\d\d\d/ .. /$end_input\d\d\d\d/){
      if ($_ =~ /(.*)\t(.*)\t(.*)/){
        print "Date  = $1" . "\t" . "Temp_moy_DegCs = $2" . "\t" . "HR
+_moy_s = $3\n";
      }
    }
  }
}

close (DATA);
[download]