The rules in the DMG don't mention this, but the ones in Complete Warrior and complete Arcane both do. You can make an argument from RAW that the PrC rules in Complete Warrior and Complete Arcane only apply to those specific books, despite the FAQ answering questions based entirely on those rules.
But as for intent, it seems pretty clear. If a character no longer meets pre-requisites, they lose prestige class abilities, but not hit dice, base saving throws, or base attack bonus, as per CWarrior, CArcane, and the FAQ. Yes, this does include spellcasting, no, this doesn't include skill points (they are part of hit dice).
However, as a GM.
You should never take away people's class abilities unless it's thematically story appropriate. If a Helm of Alignment changes the alignment of an Assassin, you should decide whether they lose their Assassin abilities or not - and be able to provide a good reason why the now Good aligned Assassin can't use his Death Attack ability on an evil beholder about to raze a town.
Similarly, if someone uses a Ring of Evasion to qualify for Fochluchan Lyrist and then later sells the Ring, you should decide whether that messes with his Lyrical abilities or not, based on circumstance, intent, so forth.
Ultimately the rule is there to stop people rorting the system. That is it's sole purpose - to stop Psychic Reformation and the like be used to get past pre-requisites. It's a heavy-handed rule, in the same way that pre-requisites are generally heavy handed and often prestige classes aren't worth the hoops that you're required to jump through - like any 'balance' rule, you should scrutinize it carefully as a GM and decide if you want it in your game.
Personally, i'd only have someone lose their class abilities as part of a plotline. I prefer to handle system rorters directly, by talking to them as a fellow human being and game player and asking them nicely to have a character with the same power level as the rest of the group. Trying to enforce the letter of the law with ill-considered and ambiguously worded rules systems like DnD is typically an exercise in failure and lack of fun - I don't recommend it, for anyone, and I definitely don't support it.