Zip library "Allowed memory size"

You can either do it like @futurewebs said. Or you will need to execute and compress those file outside of PHP with exec().
@futurewebs will be executed and will suffer the browser, server (apache/nginx) and/or PHP timeout. If you execute it outside PHP it will run on the server. But you will then need to look for said file at X interval (with Ajax) and see if it's done*.

* Should be last timestamp, haven't made a function like that in a while. Or you need to log tar/zip to a file and look for the "done" message there.