The short answer is no, there is nothing in the Apache licence which could require your website source code to be also made available under the Apache license.
Like any free software license, the Apache License allows the user of the software the freedom to use the software for any purpose, to distribute it, to modify it, and to distribute modified versions of the software, under the terms of the license.
The Apache License is permissive, so it does not require modified versions of the software to be distributed using the same license (unlike copyleft licenses - see comparison). In every licensed file, any original copyright, patent, trademark, and attribution notices in redistributed code must be preserved (excluding notices that do not pertain to any part of the derivative works); and, in every licensed file changed, a notification must be added stating that changes have been made to that file.
And from the Apache License and Distribution FAQ:
I'm not a lawyer. What does it all MEAN?
Describing legal documents in non-legalese is fraught with potential for misinterpretation. Notwithstanding the text that follows, the actual text of the license itself is legally binding and authoritative.
That said, here's what the Apache license says in layman's terms:
It allows you to:
freely download and use Apache software, in whole or in part, for personal, company internal, or commercial purposes;
use Apache software in packages or distributions that you create.
It forbids you to:
redistribute any piece of Apache-originated software without proper attribution;
use any marks owned by The Apache Software Foundation in any way that might state or imply that the Foundation endorses your distribution;
use any marks owned by The Apache Software Foundation in any way that might state or imply that you created the Apache software in question.
It requires you to:
include a copy of the license in any redistribution you may make that includes Apache software;
provide clear attribution to The Apache Software Foundation for any distributions that include Apache software.
It does not require you to:
include the source of the Apache software itself, or of any modifications you may have made to it, in any redistribution you may assemble that includes it;
submit changes that you make to the software back to the Apache Software Foundation (though such feedback is encouraged).