You don't do any error checking and you don't use warnings which would allow Perl to alert you of unopened files. When opening files, use the following:

...
open (FILE1,$p)
    or die "Couldn't open file '$p': $!";
...
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Also, your approach will be very (very) slow, because for each entry in the first file you will go through all lines of the second file. It will likely be better to store the entries from the smaller file in a hash and then go through the larger file once and compare the entries to it.

Barewords for file handles? Two arg form of open()? How about some modern perl:

open my $ID_FILE, '<', 'ids.txt'
    or die "Couldn't open ids.txt: $!";
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You forgot the smiley

use strict;
use warnings;
#this script is to match the two files and print the matched content i
++n the two files
open (FILE1,"<A") or die "Can't Open $!";

my $var=<FILE1>;
my @array=split(' ',$var);
open (FILE2,"<B") or die "Can't Open $!";
my @new;
while(<FILE2>)
{
  chomp;
  push(@new, $_);
}

my $i;
my $line;
open(OUT,">nrset.txt"); 
foreach $line (@array)
{
chomp $line;
for($i=0;$i<$#new;$i++)
{
   chomp $new[$i];
   if ($new[$i]=~/$line/i)
      {
 print OUT "$new[$i]\n$new[$i+1]\n";
      }
}
}
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Caveats:

1. For FILE A, it is unclear if you have multiple ID strings on one line, or one per line over multiple lines. I'll make this solution work with either.
2. For FILE B, are your lines "key <whitespace> letters" or "key <newline> letters <newline>", ie key and values on separate lines?

Solution:

With this solution, we look at each line of fileA and fileB exactly once, and we use a hash lookup on IDs which is fast. This reduces our complexity from O(n^2)+ from the previous solution to something closer to O(n log n), possibly close to O(n) if we're lucky with our ID hashing.

The Code

#!/usr/bin/perl

use warnings; use strict;

# open filea and parse all id strings. 
# Add id strings as keys to %wanted array.
my %wanted;
{
    open my $file, '<', "filea" 
      || die "failed to open filea : $!";
    while( <$file>)
    {
        chomp;
        @ids = split( /\s+/, $_);
        $wanted{ $_ }++ for @ids;
    }
    close $file;
}

#read fileb, parse lines of the form "id <whitespace> letters" 
#and print lines that match the id strings from filea.
{
    open my $file, '<', 'fileb'
      || die "failed to open fileb : $!";
    while (<$file>)
    {
        chomp;
        my ($id, $letters) = split( /\s+/, $_);
        print "$id\t$letters\n" if $wanted{$id};
    }
}

#OR

#read fileb, parse lines of the form "id <newline> letters" 
#and print lines that match the id strings from filea.
{
    open my $file, '<', 'fileb'
      || die "failed to open fileb : $!";
    while (<$file>)
    {
        my $id = $_;
        my $letters = <$file>;
        chomp($id);
        chomp($letters);
        
        print "$id\t$letters\n" if $wanted{$id};
    }
}
__END__
FileA:
1DWK 2RFK 
4ERH

FileB:
1DWK HRSDKKDAHJKLSDLDLLJDGHDFJJE
4ERH DFSKFHADFSBVHFWIHFWJBFS
2RFK DADUHRQWERKBNJAIJDLAJDKAKDNAKDJKSADJKAHDJASHRWEUB

FileB (alternate):
1DWK
HRSDKKDAHJKLSDLDLLJDGHDFJJE
4ERH
DFSKFHADFSBVHFWIHFWJBFS
2RFK
DADUHRQWERKBNJAIJDLAJDKAKDNAKDJKSADJKAHDJASHRWEUB
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@array = split / /, <FILE1>; # reads only one line!
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@array = <FILE1>;
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