[Update: GrandFather beat me to it.]

This might be a good place to make use of the flip-flop operator. It can help you avoid the ugly $start / $end / $started variables to keep track of where you are:

use strict;
use warnings;

while (<DATA>){

    # Returns true after matching "LV Name",
    # and returns false after
    # matching "--- Logical extents ---".
    if (/^LV Name/ .. /--- Logical extents ---/){
        print "$_";
    }


}

__DATA__
Line above
--- Logical volumes ---
LV Name                     /dev/vg00/lvol3
VG Name                     /dev/vg00
LV Permission               read/write
...
   --- Logical extents ---
Line below
[download]

Output:

LV Name                     /dev/vg00/lvol3
VG Name                     /dev/vg00
LV Permission               read/write
...
   --- Logical extents ---
[download]