PlayMoney.com

Places to invest in stocks using play money instead of real money:

Wee Seed offers play money accounts and ability to create and join groups

Young Money stock market game using play money

Most casino sites offer gambling with play money rather than real money.  However, the purpose is normally to attract users to paid sites and/or sell their information for marketing lists for other gambling sites.  Some sites, such as King Chips, sell chips to be used in play money games.