I figured out the source of all my Reaper note dropping problems in Kontakt and Vienna Instruments. I assume other sampler VSTs would be affected the same way.
One was Reaper's "Inform plugins of offline rendering state" setting. Turning it on allows me to render at high speed with no dropped notes.
Another was a bug in Reaper that I discovered by mistake. Reaper's 64-to-32-bit bridge ignores the "Inform plugins of offline rendering state" setting. Using a 64-bit instance of a VST in 64-bit Reaper works just fine. If you have any old 32-bit plugins, it's good to be aware of this bug, however.
The last has nothing to do with Reaper, but was the most frustrating to track down. In a nutshell: modern Western Digital hard drives, even fast drives like Caviar Blacks, go to sleep after only eight seconds of inactivity. It takes them about 500 milliseconds to wake back up, but the VST samplers I tested have buffers around 400 milliseconds long.
So in "online" mode, when the sampler VSTs were operating at real-time speeds, I got dropped notes. In "offline" mode, with "Inform plugins of offline rendering state" enabled, things where fine because the VSTs were able to wait for the hard drive to spin back up.
Here's more detail about how to identify and solve the problem. If you have a Western Digital hard drive, I hope this saves you some grief:
Does Your Western Digital Hard Drive Drop Notes?