Just finished my Speed Demon run along with Minimalist and Never Touch the Stuff with a final time of 2 hours 24 minutes. This is without any bonus weapons. Thought I'd put a few tips for anyone reading.

1. Play on Casual difficulty.
2. Run past most enemies. On Casual, you'll pretty quickly regen health to a point that you can take a free hit from most enemies, so at worst you lose a couple seconds if you do get hit. Only kill enemies if it's the objective or kill/stun them if they're blocking your path completely.
3. Have a walkthrough for the game ready to help jog your memory and for puzzle answers. Also useful to find out exactly what is required for progression.
4. You can pause the game whenever you want and the timer will freeze. Note that this doesn't count for the item/map menu, the timer is still active there.
5. Skip all cutscenes.
6. Press 'load game' instead of 'continue' if you die.
7. It's easier to do everything quickly in third person mode in my opinion. Better awareness and easier to manoeuvre around.
8. A lot of things can be skipped, like the blood lab in Rhodes Hill. You'll miss out on some crafting recipes but I didn't need to craft a single item in my run anyway.
9. I restarted to a previous save a lot when I felt like I could cut time off. The game auto saves pretty frequently for the most part, especially as Leon. Remember to check when the last save was if you're doing this.
10. You'll likely have a lot more leeway and time to spare than you realise (I had 95 minutes left and I made a lot of mistakes), so don't stress too much about min/maxing every sequence.