I'm moving my web application to another server and in the next few days I'll refresh the DNS to point to the new IP location.
Unfortunately some browsers and SOs keep a DNS cache that will make users point to the old IP location. Some users are rookies and they'll not refresh the DNS cache manually and I know we'll lose a lot of them in the first weeks after this change.
Is there anyway to force this DNS cache to refresh so it'll be transparent for our final users?
As far as I know, a forced update like this is not directly possible. You might be able to reduce the DNS downtime by reducing the TTL (Time-To-Live) value of the entries before changing them, if your name server service provider allows that.
Here's a guide for less painful DNS changes.
A fair warning, though - not all name servers between your client and the authoritative (origin) name server will enforce your TTL, they might have their own caching time.