What is the best method of removing weeds from between the bricks of a patio? We have a dog and don't want to use toxic chemicals if possible. The bricks are closely spaced, probably only about 1/10 inch between them.
Your best bet would be a propane fueled flame weeder — something like this. Since the surrounding material (brick) is non-flammable, you can easily direct the flame in the cracks/gaps and burn the weeds out. This is a very effective method of getting rid of weeds from landscaping stones. See also this answer on using the flame weeder.
I spent 6 years as a stone worker installing and repairing numerous types of stone patios. The best solution I have used on numerous client's stone installations to inhibit weed growth is to MANUALLY scrape them and the topmost layer of soil/dirt that has collected within the cracks. Removing all of the jointing sand from between them down about 1/4 - 1/2 inch. This is a time consuming task, but one that prepares for the best "next step"
Polymeric sand is a polymer-based jointing agent that replaces the typical sand/stone dust finish coat. Though stonedust and sand work as a great binding agent for the stone/brick, it also is easy for roots of weeds to work into.
Polymeric sand is swept into the cracks of the patio area as a finish coat, and then a GENTLE LIGHT SPRINKLE of water is applied to it to "bond" the fine particles together into a spongey, rubber like jointing material. It sheds water beautifully after is sets up, and inhibits weed growth for nearly two seasons.
A reapply is needed, but at a 1/3 of the rate of sand/stonedust solutions. And best of all... it is not a chemical.
Back when I lived in the city, my neighbour used boiling water for this. It looks rather odd to be outside with a kettle, but she swore by it. You want to be careful not to spill or splash on your legs or feet, but it's probably safer than the flamethrower and you don't need to buy or rent anything. I suspect it's easier on the concrete and bricks than salt or vinegar, also.
On our rural property we have basically set it up so this isn't a problem, but we do have a patio with "tiles" 16" on a side. If anything grows between them we just pull it up by hand.