It's more common for older models to come with Java, which will allow you to run applications based on that platform.
I have only tested on my own Samsung B2700, which is a relatively new dumb-phone, other phones may provide different results.
Testing my own phone:
While it is hard to judge the viability from my simple test I have no doubt that a simple Tetris clone or similar is at least possible, and it would probably play reasonably well.
I made the following page to test keyboard input, it is blank when loaded, but should fill basic information about keyboard events on the screen as the keys are pressed: http://ebusiness.hopto.org/phonetest.htm
Definitely it just depends on what version of html your game is written in and what your phones browser/interpreter will support