Running KeePass (or any other application) as administrator changes the integrity level. Objects of the same integrity level can potentially interact and modify each other, but an ojbect of a lower integrity level cannot modify an object of a higher integrity level. If you start your browser and KeePass both as a normal user, the integrity level of both will be set to "medium". If you start KeePass as an administrator, however, its integrity level will be set to "high" and its resources will be protected from processes that have lower integrity levels.
So, does this mean you should run KeePass as as administrator? To some degree it depends on your threat model. The general wisdom would hold that you shouldn't run with more privileges than are necessary, and if an attacker has the ability to arbitrarily modify running processes on your machine, you've already lost anyway. Unless you have very specific threats you face and you can articulate the reasons these threats put you at greater risk by following the conventional wisdom than by going down an alternate path, I would say that no, there is not such a significant security benefit to choosing to run as an administrator instead of a normal user that it would clearly justify it.