After the first time through, you're at the end of FILE, so the while() is never entered (i.e., it gets EOF right off). Try using seek.

However, it will probably more efficient (if the file is rather small relative to available memory anyway) to load in the data and then iterate over it pre-parsed, than to re-parse the file and each line every time.

At first time, when inner loop execution is over the file pointer will be in EOF. so in next time of checking While(<FILE>) this condition will get false. so it wont go inside inner while loop.

you can do any one of the following :
1. Before inner loop begins you can open file handle and at the end of inner while loop close the file handle.
2.In the beginning of the inner while loop you can make the file pointer to point to the beginning of file using 'seek'. perldoc -f seek.

Probably you would be better to construct a regular expression using Regexp::Assemble from the hash keys and perform a single pass through the file. Consider:

use strict;
use warnings;
use Regexp::Assemble;

my %hash_table = (12345 => 1, 8976 => 2, 1046 => 3);
my $assemble = Regexp::Assemble->new (flags => 'i');

$assemble->add (keys %hash_table);

my $match = $assemble->re ();

while (<DATA>) {
    chomp;

    my ($objid, $other) = split;

    if ($objid =~ /($match)/) {
        print "$1 is present in $objid\n";
    }

}

__DATA__
012345998 line matches
000104600 line matches
012389710 line doesn't match
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Prints:

12345 is present in 012345998
1046 is present in 000104600
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If that is unpractical read one line and iterate over the hash for each line. I.e. switch the two loops. This can be costly as well if the hash is big.

    seek(FILE,SEEK_SET,0);
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But this way vou have to read the big file often. As supposed above it would be better, to switch the while loops.

untested:

while(<FILE>)
{
    chomp;
    my ($objid,$other) = split(/\s+/,$_);

    while(my ($key,$value)= each (%hash_table))
    {
        if($objid =~ /$key/i)
        {
            print "$key is present in $objid\n";
        }
        else
        {
            next;
        }    
    }
}
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