Lost Temple: Players ignore all tech building requirements and don’t need heroes to cast spells.
Free Speech: Silence an opponent. (Their heroes can’t cast spells and lose all abilities until after that opponent’s next turn.)
The wording here seems like it could go either way. On the one hand, the help text on Free Speech claims that it only affects heroes, so the player would be fine if they have any non-hero method of casting spells. On the other hand, Free Speech says that it silences the opponent, which implies that it would still prevent that opponent from casting spells.
The rulings don’t seem to give any definitive answer either. No rulings exist for Lost Temple (or any map card as far as I know), but here are the most relevant rulings on Free Speech:
If you use Free Speech on an opponent and that opponent plays a new hero on their next turn (in which they are still silenced), that new hero loses all abilities and can’t cast spells. — Sirlin
A silenced opponent can still play spells using Cinderblast Dragon or Guargum, Eternal Sentinel — sharpo
The first one implies that the effect of silence is applied to the player, since it affects heroes who aren’t even in play yet, whereas the second one (which you referenced in your question) introduces an exception, setting a precedent that could potentially allow for more exceptions. Thus, I conclude that if this is going to be relevant to you in a real game, you’ll need to tag sharpo so he can clarify.
For what it’s worth, I would be inclined to say that Free Speech would still prevent spells from working in the Lost Temple (unless you had a spellcasting unit). Free Speech does say that it silences an opponent, and there have already been discussions about how the text on the cards was written without accounting for the map cards. For example, Rich Earth technically reduces the cost of workers by 1, but that’s functionally identical to making them free unless a specific map is being used, so they wrote the card’s text to say “free” for simplicity. This looks like a similar case to me.