#!/usr/bin/perl

use strict;
use warnings;
use Data::Dumper;

my $qfn1 = "File1.txt";
my $qfn2 = "File2.txt";

my %positions;
{
   open(my $fh, '<', $qfn1)
      or die("Cannot open file \"$qfn1\": $!\n");

 while ( <$fh> ) {
    chomp;
    my ( $key, $pos ) = split /\s+/;

    
    push @{ $positions{$key} }, $pos;
}
}
print Data::Dumper->Dumpxs( [ \ %positions ], [ qw{ *positions } ] );
my %sequences;
{
   open(my $fh, '<', $qfn2)
      or die("Cannot open file \"$qfn2\": $!\n");

   my $key;
   while (<$fh>) {
       if ( s/^>// ) {
           $key = ( split /\|/ )[1];
       }
       else {
           chomp;
           $sequences{$key} .= $_;
       }
   }
}
for my $key (sort keys %positions)
            {
               for my $key (keys %{$positions{$key}})
               {
   foreach my $value (@) {
    if (! exists( $sequences{$key} )) {
        warn "KEY $key in File1 not found in File2\n";
        next;
    }
    if ( length( $sequences{$key} ) < $positions{$key} ) {
        warn "KEY $key: File2 string length too short for File1 positi
+on value\n";
        next;
    }
     my $index = rindex( $sequences{$key}, "ATG", $positions{$key} );
    if ( $index < 0 ) {
        warn sprintf( "KEY %s: No ATG in File2 string prior to positio
+n %d\n",
                      $key, $positions{$key} );
        next;
    }
    $index += 3 while ( ($index + 3) < $positions{$key} );
    print "$key $positions{$key} " . substr($sequences{$key}, $index, 
+3) . "\n";
}
    }
}
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Please check your script for syntax errors:

$ perl -c script.pl
syntax error at /tmp/test.pl line 45, near "if"
syntax error at /tmp/test.pl line 61, near "}"
$
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some comments upon your code.

• whole code: a clear indentation would help reading and debugging the code! perltidy can help with this!
• line 31: why do you use s/// here? m// should do the job! The replacement of > (at the beginning of the string) is not needed here as you split the string and use the second part. So why do you remove the >?
• line 42: %positions is a Hash-of-Arrays; why are you trying to dereference it as a Hash-of-Hashes?
• line 44: what's that single @? Maybe you want to dereference the array reference in $positions{$key}? If so, you still should rename $value to $position!
• What do you think $value is for, as you don't use it in your for-loop.
• line 49: why are you still comparing a string length with an array reference?
• I stopped analyzing the following lines. But it looks like as you are not aware about what you have stored in $positions{$key}

As tilly already mentioned, you should start learning about perl's complex data structures. Recommended documentation:

• perlreftut
• perldsc

Update:

• comment on line 42 added.