Your code as best I can make it out: (please use code tags in the future)

#!perl 
$program = "c:\\progdir\\bin\\proggie.exe"; 
$dstfile = "d:\\scripts\\log\\archive\\proggie.log"; 
$arg0 = "\"Blah.Blech.>\""; 
$timeout = 60; 
$starttime = time; 
open( PRG, "$program $arg0 |") || die "Couldn't open PRG: $!\n";
open (LF, ">$finaldest"); 
while (<PRG>) 
{ 
  print LF "$_"; 
  $difftime = time - $starttime; 
  last if ($difftime > $timeout); 
} 
close (LF); 
close (PRG);
[download]

My untested suggestion:

#!perl 
use strict; # ALWAYS! (Why? Helps catch typos before run time... among
+ other things)
$|=1; # This disables buffering. JCWren's Suggestion.
my $program = 'c:\progdir\bin\proggie.exe'; 
my $dstfile = 'd:\scripts\log\archive\proggie.log'; 
my $arg0 = '"Blah.Blech.>"'; 
my $timeout = 60; 
open( PRG, "$program $arg0 |") or die "Couldn't open '$program $arg0',
+ $!";
open(LF, ">$finaldest") or die $!; 
my $inloop = 0;
my $endtime = time + $timeout; # Only do the math once.
while( <PRG> and $endtime >= time  ) # '>=' or '>', your choice.
{ 
  if( not $inloop )
  {
    # This will get run the first time, but not after.
    # As per JCWren's Suggestion.
    print STDERR "Inside Loop\n";
    $inloop = 1;
  }
  chomp;
  print LF $_, "\n"; 
} 
close(LF) or die $!; 
close(PRG) or die $!;
[download]

Of course most of this is clean up, I can't see anything really wrong with your code. I don't usually use last since I don't always see it when looking for exits from a loop. Of course, last ought to work, so I tried a simple cmd liner:

C:\>perl -we "$i=0;$stop=time +2;while(1){print $i++, qq!\n!; last if 
+time > $stop}"
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Which ran for roughly two seconds, no problem.