I had a game for my Apple//GS way back, but I was too young to understand it. I am half inclined to purchase Shadowgate from GOG, but I wanted to ask a question first - is there a time limit with Shadowgate? If I am idle in game, will the game time eventually run out?

Yes. No. Time is "turn based" meaning only when you do something in game it counts.

Keep in mind this is not exactly like original game from Apple. It's modern remake or could even say re-imagination of that game.

Edit: There is one playmode where torches don't burn out (basically no time limit) but only for lowest difficulty which means you'd miss most puzzles (and some locations even) so kinda sucks.