First of all, in Tk (or any gui program) using "sleep" will block the gui, so it's a "no-no". But to show you how to make it work ,in terms of your question, you need to set $|=1 and update your text widget after each print. Like:

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

$|=1;

my $mw = tkinit;

# a frame in my Tk main window 
my $frame = $mw->Frame;

# a text widget with attached scrollbars in my frame 
my $OutputText = $frame->Scrolled('Text',
   -height => '10',
      -width => '50',
         -scrollbars => 'osoe'
         );

         # pack frame and text widget 
         $frame->pack(qw/-side left -fill y/);
         $OutputText->pack(qw/-side bottom -fill both -expand 1/);

         # here I try to tie STDOUT to the text widget 
         my $widget = $OutputText->Subwidget("text");
         tie *STDOUT, ref $widget, $widget;

$mw->Button(-text=>'start', -command => [\&start])->pack;;

MainLoop;

sub start{

for ( 1..10 ) {
   sleep 1;
      print "foo\n";
      $OutputText->update;
      }

}
__END__
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But here is probably a better way, without using sleep. The reason is that a repeat loop, lets the Tk event loop proceed, and respond to the gui, while sleep will block everything until it is done.

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

use Tk;

my $mw = MainWindow->new();
my $tx = $mw->Text()->pack();

tie *STDOUT, 'Tk::Text', $tx;

$mw->repeat(1000, \&tick);

MainLoop;

my $count;
sub tick {
 ++$count;
 print "$count\n";
}
[download]

However, you don't say what you are ultimately trying to do, and tie'ing STDOUT in Tk is not done very often, because there are usually better ways to accomplish it, like printing directly to the text box. What kind of program are you trying to make?