How do you clear out all old query plans from within Microsoft SQL Server?
You asked two questions here. First, you want to know if you can remove all plans stored in memory for an instance of SQL. That is done with DBCC FREEPROCCACHE as Matt M suggested.

The second question you asked is "How can I do this without impacting 30 or so users that rely on other databases on the same server?". The short answer is "you can't". If you remove all plans than the other users that are relying on plans being in memory will likely suffer a performance hit.

The workaround for this requires some manual intervention. You can use DBCC FREEPROCCACHE to remove specific plans providing you have the plan_handle. 

From what you are describing above it does sound like a plan issue, but I am not certain that removing plans is the answer. I would point you in the direction of parameter sniffing before thinking about removing plans:

http://blogs.msdn.com/b/conor_cunningham_msft/archive/2010/08/11/conor-vs-misbehaving-parameterized-queries-optimize-for-hints.aspx

You should be able to optimize the queries rather than fooling around with DBCC FREEPROCCACHE on a scheduled basis. I would also advise that you spend time analyzing the wait events for your instance.