[Update #2]: Resident Evil 3 and its companion Resident Evil 3 Resistance have been released, though sadly they don't seem to have lived up to the high standards set by Resident Evil 2.
[Update]: The demo has finally arrived, and I'm happy to say Resident Evil 3 is looking pretty darn good right now!
A little while ago Capcom announced that Resident Evil 3 will not only be getting a playable demo, but that it will be coming in the "near future". They certainly weren't joking about that last part as the demo has now been slated to launch this March 19 on Steam, PlayStation 4 and Xbox One!
What exactly the demo will contain, and how long it will be, Capcom has unfortunately not clarified. However, given that Resident Evil 2's demo ended up being a massive success, I can only imagine that Resident Evil 3 will follow in the same footsteps. In other words, don't be surprised if its ends up being a 30-minute demo showcasing the introductory portions of the game, all topped off with a brand new trailer at the very end.
If you're in the mood for something a little bit different, it's also worth mentioning that Resident Evil 3 will come bundled with Resident Evil Resistance - an asymmetrical 4v1 multiplayer game that has one player take on the role of the evil mastermind in order to crush the survivors before they can complete their objectives and escape. A fairly simple formula, but as games like Dead by Daylight have shown, it can be pretty good fun.
However, while Resident Evil 3 will only have a short demo, Resident Evil Resistance will have a fully fledged Open Beta this March 27th. It will end once the game launches on April 3rd, which means that the gameplay will likely be pretty polished, so it should give you a solid impression of just what Resistance is all about, and more importantly, if it's even worth your time and effort.
Have fun with the games once they release, and I'll leave you with the recently posted and rather brief teaser trailer: