Full error checking and safer argument handling:

open(my $fh, '-|', 'gunzip', '-c', '--', $input_file)
   or die("Can't start gunzip: $!\n");

while (!eof($fh)) {
   defined($_ = <$fh>)
      or die("Error reading from gunzip: $!\n");

   ...
}

close($fh)
   or die("Can't close gunzip: $!\n");

($? & 0x7F) and die("gunzip died from signal ", ($? & 0x7F), "\n");
($? >> 8)   and die("gunzip exited with error ", ($? >> 8), "\n");
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close

Hmm... didn't quite trap at the open statement.

>./errorcheck.pl notexists.gz
gzip: notexists.gz: No such file or directory
Can't close gunzip:
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I guess the "Can't close gunzip" occurred due to a broken pipe?

Of course it didn't trap at the open statement. The error hasn't happened yet. As for the incorrect error message, the code should be:

...

if (!close($fh)) {
   $!          and die("Can't close gunzip: $!\n");
   ($? & 0x7F) and die("gunzip died from signal ", ($? & 0x7F), "\n");
   ($? >> 8)   and die("gunzip exited with error ", ($? >> 8), "\n");
}
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gzip: notexists.gz: No such file or directory
gunzip exited with error 1
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Can you suggest a way to have open fail when the pipe command has non-zero result code?

Simple way (that avoids all the "safer argument handling" bullshit) :

open my $fh, "gunzip -c $input_file |" and !eof($fh) or die "$!/$?";
while(<$fh>){
    ...
}
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Elaborate the error handling to suite your needs.