NetWalla identified your immediate problem, but there is an issue with your regex. The spaces around third and four are required to match spaces in the string you are matching against. On top of that, you don't prevent matching words that contain third or four (fourth for example). You can fix both issues like this:

use strict;
use warnings;

local $/ = "\n\n";

while (<DATA>) {
    if (/\b(third|four)\b/si) {
        chomp;
        print "line is: $_\n";
    }
}

__DATA__
This is line one.
Line Two is this.
Third line starts here.
This is line four.

This is line five going fourth.
This is line six.
This is the seventh line.
This is line eight.
[download]

Prints:

line is: This is line one.
Line Two is this.
Third line starts here.
This is line four.
[download]

the \b matches word boundaries.