The trick is pulling the dollar bill out as quickly as possible. Too slow and there's too much friction between the bottle and dollar bill, causing the bottle to tip over. My daughters weren't able to pull the dollar bill out fast enough, but I finally got it on the try in the video above. It was cool seeing how long the bottle wobbled before settling back to a rest.
If you have bottles of different sizes you can test which bottles work better. Tall? Short? Fat? Skinny? What about a full bottle? Does a full bottle work better than an empty bottle? If you use a full bottle, you may want a towel ready to go in case the bottle breaks.