SQLite doesn't support any kind of concurrency, so you may have problems running it on a production website. If you're looking for a 'lighter' database, perhaps consider trying a contemporary object-document store like CouchDB.
By all means continue to develop against SQLlite, and you're probably fine to use it initially. If you find your application has more users down the track, you're going to want to transition to postgres or mysql however.
The author of SQLlite addresses this on the website:
SQLite usually will work great as the database engine for low to medium traffic websites (which is to say, 99.9% of all websites). The amount of web traffic that SQLite can handle depends, of course, on how heavily the website uses its database. Generally speaking, any site that gets fewer than 100K hits/day should work fine with SQLite. The 100K hits/day figure is a conservative estimate, not a hard upper bound. SQLite has been demonstrated to work with 10 times that amount of traffic.
SQLite will normally work fine as the database backend to a website. But if you website is so busy that you are thinking of splitting the database component off onto a separate machine, then you should definitely consider using an enterprise-class client/server database engine instead of SQLite.
So I think the long and short of it is, go for it, and if it's not working well for you, making the transition to an enterprise class database is fairly trivial anyway. Do take care over your scheme however, and design your database with growth and efficiency in mind.
Here's a thread with some more independent comments around using SQLite for a production web application. It sounds like it has been used with some mixed results.
From personal experience, I remember back in the dark days of web development using MS Access and Filemaker Pro that file locking was a persistent and painful problem. SQLlite may handle this more elegantly, but essentially it's the same problem, and I doubt much would have changed.