The range operator (..) might do what you're looking for. For details, see Range Operators operator, range range .. ...

while (<DATA>) {
    if ( m{<sb>} .. m{</sb>} ) {
        s/<s1>/<h1>/g;
    }
    else {
        s/<s1>/<section1>/g;
    }
    print;
}
__DATA__
<s1>King of England
JAMES, by the Grace of God, King of England, Scotland, France and Irel
+and, Defender of the Faith...
...
This first selection was written in England to establish the colony of
+ Virginia
<sb><s1>Rights of Landowners
We, greatly commending, and graciously accepting of, their Desires for
+ the Furtherance</sb>
<s1>King of Perl
This is only a test.
<s1>King of Operators<sb><s1>Well, maybe not</sb>
[download]

Output:

<section1>King of England
JAMES, by the Grace of God, King of England, Scotland, France and Irel
+and, Defender of the Faith...
...
This first selection was written in England to establish the colony of
+ Virginia
<sb><h1>Rights of Landowners
We, greatly commending, and graciously accepting of, their Desires for
+ the Furtherance</sb>
<section1>King of Perl
This is only a test.
<h1>King of Operators<sb><h1>Well, maybe not</sb>
[download]

My only concern is what happens if <sb> and </sb> appear on a line together or on a line with <s1> outside of them (see the last line of my example). Your sample doesn't have that case present, but your real text might. If so, you'll have to parse the text more deeply or preprocess it to remove the "problem" areas.

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