I know that DNN can run successfully on a VPS system, in fact I ran 5 installations of DNN side-by-side on a VPS for 8 months without any issues. These sites were running about 1200-1500 visitors per day total.
The key here is to validate a few things first of all with server configuration and DNN Configuration
1. How much ram do YOU have allocated, this needs to be at least 1 GB at minimum
2. Are you running SQL Server on the same box? If so, did you remember to setup the memory limit? If not, this will be a HUGE issue
3. DotNetNuke configuration, have you modified the frequency of the expensive scheduled tasks, such as the Search Engine Scheduler.
4. How large is your Event Log, clear it if it is large.
Once you have validated a few of these hopefully you have either identified the issue, or at least discounted a few things to help move forward.
If you have questions about any of the above feel free to drop me an email, msellers@iowacomputergurus.com, I do DNN performance optimizations all the time.