If you really wanted to do it with a single regular expression you'd need to slurp the file, and that's not recommended because it doesn't scale well. However, the flip flop operator (scalar context range operator - ..) provides an interesting solution:

use warnings;
use strict;

while (<DATA>) {
    print if s/{AUTHOR}\n// .. (s/{AUTHOR}\n//, s/{(?!AUTHOR})\w+}\n//
+);
}

__DATA__
{TAG}
tag1
{AUTHOR}
By June Fletcher
JOURNAL
{TAG}
tag2
TOM MacCUBBIN
{DATA}
data1
{AUTHOR}
Richard White
{AUTHOR}
MacCUBBIN
{SOUR}
[download]

Prints:

By June Fletcher
JOURNAL
Richard White
MacCUBBIN
[download]