When you run your site in a live mode, you can still have the publishing restictions such as publish dates for items, do not publish flag and lifetime for versions to remain in effect. All you need to do is to find your site in <sites> section of web.config file, and add filterItems="true" to it.

The attribute is actually described in inline help above the sites section, but its often missed. Turning it on makes running some sites in live mode much more feasible. The catch that still remains (for live mode) is that if you use html cache, it will not be cleared because there's no publishing event to trigger that.