Short answer is no. If you're making a game with simple to moderate complexity you really don't need to worry about threads. As mentioned before, modern processors are fast, and most of your cost is probably going to be on the rendering side.

Long answer is yes. You may not need threads for strictly performance reasons, but adding parallel capabilities into your engine will make it flexible if you want to add features that may require it (networking, costly calculations, file I/O, etc). You don't have to go crazy and thread everything, but always take a step back and say "Could these things run at the same time with minimal effort?"

Don't be discourage from giving it a try. Threaded game engines aren't necessarily complex. It all depends on you want to multithread / how granular your threaded data is.

The question you need to ask yourself is: "are you developing a game, or are you developing an engine?" (Doesn't have to be a binary answer, but you should definitely be aware of your place on the spectrum).

If you're developing a game, then the answer is no. If your game is fairly straightforward, then more threads will only get in the way of the game development with no real added benefit.

If you're building an engine, then the answer is yes. You may not need threads for strictly performance reasons, but adding parallel capabilities into your engine will make it flexible if you want to add features that may require it (networking, costly calculations, file I/O, etc). You don't have to go crazy and thread everything, but always take a step back and say "Could these things run at the same time with minimal effort?" The game will only suffer in the sense that you'll have to spend more effort into feature implementation rather than game play code.

Either way, don't be discourage from giving it a try. Threaded game engines aren't necessarily complex. It all depends on you want to multithread / how granular your threaded data is.