It's only doing what you ask it to do:

$text = $text . $1 . "\n";

Since $text is effectively a global variable, it will always get longer. If you were to do something more like this, you wouldn't have this trouble:

while (1) {
    # get webpage

    my $text;
    while (<TEXT>) {
        if (/src=([^>]*)>*$/i) {
            $text .= "$1\n";
        }

    # continue as normal

    sleep 30;
    }   
}
[download]

This ensures that $text is cleared on every iteration of the outer loop.

Stylewise, there are more Perlish ways to grab webpages. You might like the LWP library, especially LWP::Simple. It really is as simple as the name says. Also, don't forget to check the results of your open calls. They may fail.

Update: Removed spurious forward whack in regex. There's no need for the anchors, greedy match anythings or the ignore whitespace and match all flags. They don't really hurt anything, though.