Do you mean the "Managed QR Code"? It configures some of settings available in the Collect app. I don't think there would be a server performance issue. But it will depend. What are your server specifications and does your workflows involve transferring very large amounts of data (e.g., a very large number of data collectors, frequently updated very large external files, etc.). I think any issues with server performance due to your data collection workflow will likely happen regardless of if you use the managed QR Code or not.
Have you reviewed the Central System Requirements notes in the documentation? "There is very very little chance it will mishandle or corrupt your data even if it is run on a weak machine and runs into extreme traffic."
There is still a chance, though, especially with form attachments that change often which is now more common with Entity Lists.
@danbjoseph asks good questions. I think it's also helpful to know whether you're using the default setup with a database run on the same VPS or whether you have your database on a separate service/machine. What kind of monitoring do you have on the database? Does it show any spikes in resource usage? I would expect that the database was CPU-bound leading to slow queries and timeouts when many data collectors were updating at once.
In fact, if I understand you correctly, using a QR-code with automatic synchronization does not affect the memory any more than a QR-Code with manual synchronization?
There are too many variables to make a blanket statement like that. Are you using entities? Are you frequently updating form attachments? How many Collect clients will be connected to your server?