Please, check out the return from pg_last_error($dbconn); after the pg_send_query() call and post the result in there.

Modified test script with requested pg_last_error output. 
-------
<?php
for($i=1;$i<=10;$i++)
{
	echo "Interation $i\n";

	$dbconn = pg_connect("user=cstdenis dbname=postgres", 
PGSQL_CONNECT_FORCE_NEW);

        // Will block in the second loop interation
	pg_send_query($dbconn, "select pg_sleep(5);");
	echo "Last error: '".pg_last_error($dbconn)."'\n";
}
?>

Output:
-------
Interation 1
Last error: ''
Interation 2
Last error: ''
Interation 3
Last error: ''
Interation 4
Last error: ''
Interation 5
Last error: ''
Interation 6
Last error: ''
Interation 7
Last error: ''
Interation 8
Last error: ''
Interation 9
Last error: ''
Interation 10
Last error: ''

This won't block:

<?php

for($i=1;$i<=10;$i++)
{
        echo "Interation $i\n";

        $dbconn[$i] = pg_connect("dbname=postgres", PGSQL_CONNECT_FORCE_NEW);

        // Will block in the second loop interation
        pg_send_query($dbconn[$i], "select pg_sleep(5)");
        echo "Last error: '".pg_last_error($dbconn[$i])."'\n";
}

?>

So the issue only happens when you "overwrite" an existing connection

Sorry, hit submit too early. By overwriting the connection variable, you force the old connection to be closed. In order to do that, PHP is waiting for the query to complete and discard its output.

It should be able to throw away the variable without closing the connection 
(much 
like calling pg_connect without assigning to a variable leaves it open). I 
realize doing this could cause a memory leak for the duration of the script, but 
as long as the implicit close at script termination still happens it should be 
fine.

In any case, please update the documentation to make this more clear since 
blocking on variable assignment is not something one normally expects to happen.