If this is an IOS router, then "show hosts" will show you the current resolver cache.
I don't know the answer. However you can be reasonably confident that the resolver wont cache an entry beyond the TTL in the DNS response. So lower the TTL in the record you are worried about on your DNS server to be equal to the time you would like the router to repeat the DNS lookups.
For example, if you make the DNS TTL of your entry 60s,then no matter what the router should look it up every 60s.
From my memory, the IOS resolver flushes the cache on a scheduled interval, so it is never exactly equal to the TTL, but it is close enough.