Anything involving moving or renaming folders with hundreds or thousands of notes in them will lock up Obsidian for several minutes.
Obsidian interprets moving a folder as creating that folder anew, so if you’re using Obsidian Sync it will delete that entire directory from the remote site and re-upload everything. Then each device synced to it will have to delete and re-download everything as well, requiring you to keep the client device on and open to Obsidian the entire time.
DO NOT WAIT. If you wait to sync large changes to a client device, this can cause errors in the sync, which can propagate back to the primary location of your Library. This can cause a persistent divergence between the state of the client device and the primary device. This is fiddly to fix.
When syncs are occurring, the client Obsidian becomes useless for taking notes. (This seems to only happen on devices with little onboard RAM like phones and tablets.)
Large Obsidian libraries do cause slowdown in selecting files (the contents of each note take noticeably longer to render), especially on devices with less memory, such as phones and tablets. It lags noticeably, and on devices with less RAM, has even caused the client to crash.
With large libraries on lower RAM, it will frequently trigger a sync in the middle of taking a note, and this can result in 5-10 words (30-60 characters) of work being lost. There is no way to predict or avoid this.
The same goes if you are copy-pasting from another app on the same device. I have lost substantial amounts of work switching away and switching back. (50 words or so.) I have gotten in the habit of triggering a manual save before switching apps, or just after finishing a note, so I don’t lose what I just typed. Apps which themselves take up a lot or all the memory, such as Kindle, are especially bad on this count.
I’m sure there are other difficulties, but these are the ones that come to mind right now.
In short: Large libraries and small amounts of RAM have a problem. Large libraries and sync has difficulties. Neither of these are particularly surprising.
Source: A couple of years experience of using Obsidian with a very large Library. Currently 86,313 files spread across 6,183 folders.