My three-year-old wants whatever my five-year-old is playing with. He will throw a tantrum if he does not get it, and we find ourselves relying on his older sister to diffuse the situation by "being a big girl" and letting him have what he wants. SHE HAS BEEN TREMENDOUS by sharing, but I hate to have to always place that burden on her. Also, I don't think he is learning a good lesson by getting what he wants. I try to wait out his tantrums but then my patience is so minimal with him these I end up yelling at him or having big sis give him what he wants. Anyone have any advice?
I suggest setting your foot down and keeping it down. You are teaching him that he can get what he wants if he just throws enough tantrums to wear down your patience. For the next six months, purpose that if he even begins to throw a tantrum, there is no way in the world he is going to get what he wants, and will probably instead be disciplined. The next time he throws a tantrum, have some negative consequences. Not only does he not get his sister's toy, he also get something else taken away, or he gets grounded to his room. This is getting worse only because you gave into it.
And this is being very unfair to your five year old daughter. What is being communicated to her is that her interests are not as important as her brother's because she does not throw tantrums.
You are in fact reinforcing bad behavior. Stop it!
I would suggest getting him more toys of his own - toys that his sister would also be interested in, so she will want to borrow them from time to time. Then make it clear that lending toys is voluntary - don't force either child to do it - and point out that if he wants to borrow his sister's toys, he needs to lend his toys to her sometimes.
Also, you may want to find a place where you can put him in timeout so you can withstand a tantrum without yelling at him. Giving in to tantrums just encourages them.