Does the Play All card require the removal of any Hand Limit rule cards currently in play?
No. Play All replaces the Play One, or any associated play rule cards.
Of the many New Rule cards in Fluxx, there are several archetypes, including: Draw, Play, and Hand Size. Each are categorically separate from the others, which is why each directs you to discard any previous associated rules.
Adding on to Drunk Cynic, they're separate because they affect specific mechanics of the game. While these mechanics may interact, they are not identical and, with the ever-changing nature of the rules, having Play All remove Hand Limits would be having one archtype rule over another.
Here's an example:
If you have a no hand limit rule in effect and a play 2 and draw 1 in effect, you would never have to worry about the hand limit anyway, but you don't discard it because, if the draw count is increased to 3, you still want that no-hand limit in effect.
Similarly, when you play the Play All rule, you still want to have the hand limit in effect so that, if Play All is eventually replaced, you have the most recent hand limit rule to reference.