There's no guaranteed way to force the user to clear the DNS cache, and it is often done by their ISP on top of their OS.
It shouldn't take more than 24 hours for the updated DNS to propagate.
Your best option is to make the transition seamless to the user by using something like mod_proxy with Apache to create a reverse proxy to your new server.
That would cause all queries to the old server to still return the proper results and after a few days you would be free to remove the reverse proxy.
If you changed the settings in a record but your traffic is still being routed to the old resource, such as a web server for your website, one possible cause is that DNS still has the previous settings cached.
Each record has a TTL (time to live) value that specifies how long, in seconds, that you want DNS resolvers to cache the information in the record, such as the IP address for a web server.
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.