The question of strength training comes up pretty regularly on the various BJJ forums around the internet and opinions are usually mixed. I think I can help clear up some of the confusion around when strength training will benefit you and when it won't.
You Should Strength Train When:
1. You are competing at the top level of your sport where every tiny edge is required to defeat people who are of equally immense skill. This applies to pretty much no one who reads my articles, and pretty much no one who is posting this question. If you are competing at the highest level then you know what you need to do to win.
2. Doing so will not take time away from your skills training. If you have the option of strength training OR going to class, you should always go to class. A lot of people who ask this question DO have free time that they can't use to train BJJ, and there is absolutely nothing wrong with hitting the weights during those times.
3. When you are at an EXTREME physical disadvantage where rapid gains are easy. Weightlifting, like most things, is a game of diminishing returns. If you have never really lifted weights, are really tiny, or really over weight, and are being consistently outpowered by people your size and smaller then lifting weights will have rapid results in relatively little time. This is one of those rare times when putting your time into a good strength training routine will bring you results very quickly and is probably worth the time investment. Very few people are weak enough or out of shape enough to qualify though.
4. You have a specific physical issue that will benefit from it. There are many medical issues that can be helped by regular strength training. You shouldn't forego that benefit just because the BJJ community starts spazzing out at you.
5. You enjoy weight training for its own sake and don't care whether your BJJ progress is slowed by trading some BJJ Classes for weight training. If you would rather spend 3 days lifting and 2 days training BJJ than the other way around, then do that and don't worry about.
6. If you are over 30. As you age you can start to lose muscle mass and bone density, both things that weight training can help with. There are serious health benefits to weight training that can not be ignored.
When you should NOT Strength Train:
1. When you feel like you only got tapped because someone was stronger than you are. This is almost never the case and hitting the weights instead of hitting the mats will only impede your progress.
2. When you are ranked lower than purple belt. At white and blue belt the diminishing returns still generally favor skill training over strength training except in extreme cases. Once you hit purple it takes longer to improve at BJJ, so hitting the weights can help lift your game a few extra steps by opening up more options for you.
3. When you have JUST started training so that you can lose weight and get in better shape. Piling weight training on top of BJJ will kill you. Or you'll quit both of them because by the fourth day of alternating BJJ and squats you won't be able to move.
4. In order to get in better shape before you start training. This will not work. Grappling is a completely different animal from any other physical activity you can engage in. Being the worlds strongest man won't stop you from throwing up all over yourself thirty minutes in to your first class.
5. If you can already bench ~1.5x your weight, do 20 pullups, squat 2x your weight, and deadlift 2.5x your weight. If you can do those things then you are going to be hitting the line of diminishing returns almost immediately when you start lifting and you'll get almost no benefit from it for your jiujitsu. You would be better off doing gymnastics or finding some people to roll with.
6. When you feel like the only reason you couldn't tap someone was because you weren't strong enough. Again, this is almost never the case and trading mat time for strength training time is just going to make things worse.
I'm sure all kinds of people will disagree with me, so feel free to leave vitriolic feedback wherever you like.
Edit: It was mentioned that many people will fall into both categories in one way or another. Usually I imagine with people being over 30 or at an extreme physical disadvantage, and below the rank of purple belt. If you are over 30, you should be weight training anyways. Regardless of other factors. There are simply too many health benefits to it for you to ignore it. However, that question has led me to think that this article might be better represented as a flow chart. So I think I will make one.