Visiting Shenzhen is highly recommended if you have any interest in electronics at all --- I'd say it's definitely an eye-opening experience, and can change your perspective on the manufacturing costs of all the things we usually take for granted.
I would love to do more factory/production tours but in many countries it seems that insurance and legal issues have made them too difficult. Brewery tours are common (the Budvar one in the Czech Republic is great) but I'd be just as interested in car manufacture, electronics, furniture, clothing, almost anything.
Are there any formal tours in China? I don't mean the sort of tour where you are driven around in a bus visiting factories, but an index of factories that are willing to take an appointment to visit for a fee, just out of curiosity.
In the UK most factory tours seem to involve either a) people related in some way by business (clients here for a meeting, why not take them round the factory), b) student groups, c) family/friends or d) the local members group for professional institutions (eg in the UK the institute of mechanical engineers). I'd imagine your best bet is to see what the latter groups are doing (although whether or not you can attend without having the right degree is probably fairly dependent on local policies and how many other people want to attend).
In the USA at least, a lot of factories can give tours if you have a friend who works there. For example, you could tour the SpaceX rocket factory in Hawthorne, CA, which is AWESOME.
I guess this is the case, as they like to wow suppliers/customers or use it as a way to keep a trucker/customer to wait X minutes until the order is actually done without complaining too much.
The "tour for a friend" is probably seen as an opportunity to practice the tour when it doesn't really matter, test new things, etc.
I got to tour a sulphuric acid plant. It was fun. The friend worked in the closest equivalent to hell that I've ever seen (well, the cauldron part anyway). I love walking by a X000 fahrenheit tank with the guide saying as you have to crouch around to get around it "Don't go near, hot!".
I'd second this. I've been there a couple times and it is quite the experience. I only visited one electronics factory, and the rest were other types of factories that make mechanical components (injection molding, die casting, sheet metal, silicone molding, etc.) but they were all fascinating and I have a lot better understanding about the supply chain now.
I'll third it too. I visited earlier this year with a UK trade mission for the Maker Faire and to meet suppliers. I visited several factories, both with the trade mission and independently, and it was one of the most fascinating weeks of my life. One place that I'd definitely suggest visiting, if only for the contrast with most other manufacturers, is Seeed Studios. Really cool place, and totally unlike any other place I visited. It's also very well set up for tours.
bunnie has organised another trip there near the end of this month, if you're interested: http://www.bunniestudios.com/blog/?p=4087