I recently read this Question about SQLite vs MySQL and the answer pointed out that SQLite doesn't scale well and the official website sort-of confirms this, however.
How scalable is SQLite and what are its upper most limits?
Answer
Yesterday I released a small site* to track your rep that used a shared SQLite database for all visitors. Unfortunately, even with the modest load that it put on my host it ran quite slowly. This is because the entire database was locked every time someone viewed the page because it contained updates/inserts. I soon switched to MySQL and while I haven't had much time to test it out, it seems much more scaleable than SQLite. I just remember slow page loads and occasionally getting a database locked error when trying to execute queries from the shell in sqlite. That said, I am running another site from SQLite just fine. The difference is that the site is static (i.e. I'm the only one that can change the database) and so it works just fine for concurrent reads. Moral of the story: only use SQLite for websites where updates to the database happen rarely (less often than every page loaded).
edit: I just realized that I may not have been fair to SQLite - I didn't index any columns in the SQLite database when I was serving it from a web page. This partially caused the slowdown I was experiencing. However, the observation of database-locking stands - if you have particularly onerous updates, SQLite performance won't match MySQL or Postgres.
another edit: Since I posted this almost 3 months ago I've had the opportunity to closely examine the scalability of SQLite, and with a few tricks it can be quite scalable. As I mentioned in my first edit, database indexes dramatically reduce query time, but this is more of a general observation about databases than it is about SQLite. However, there is another trick you can use to speed up SQLite: transactions. Whenever you have to do multiple database writes, put them inside a transaction. Instead of writing to (and locking) the file each and every time a write query is issued, the write will only happen once when the transaction completes.
The site that I mention I released in the first paragraph has been switched back to SQLite, and it's running quite smoothly once I tuned my code in a few places.
* the site is no longer available