Many of this facts and myths were compiled by NCDC.
"When a tornado warning is issued, you should open all the windows in the house."
A common tornado myth is that opening the windows will equalize the pressure in your house, which is thought to protect your home from damage. This is totally unnecessary and wastes valuable time in getting to your storm shelter location. If a tornado is going to pass close enough to do damage to your house, there's nothing you can do to minimize it, and making the effort is only risking your life.
"Tornadoes do not hit big cities."
Tornadoes hit large metropolitan areas with a high relative frequency. For example, in 1999, an F-5 tornado tore through Oklahoma City, and in 2007, an F-2 tornado crossed through Atlanta, Georgia. More recently, an incredibly destructive F-4 tornado traveled from Tuscaloosa to Birmingham, Alabama in the April 25-28 outbreak. Tornadoes are not diverted by any structure or terrain. Tornado strikes on large cities seem less common only because there are few cities relative to the size of the rural areas in the U.S.
"Tornadoes have picked up objects and people and set them down without damage or injury."
There are multiple reports of objects, animals, and people being transported up to a quarter mile without serious damage or injury. In April 2011, a young boy was in bed and was sucked from his home by a large tornado and set down a few hundred yards away. After the tornado had passed, the boy walked back to his home and family without injury. While these stories are true, they are incredibly rare. You should never assume you're going to walk away from a tornado if you don't seek shelter.
"Tornadoes don't happen in the mountains"
Tornadoes are not diverted by any structure or terrain. Tornadoes have been documented in the mountains, including damage from an F-3 above 10,000 feet.
"Tornadoes won't cross over rivers or other bodies of water."
It doesn't matter what's in front of it - a tornado will pass over or through it. For example, the Natchez, Mississippi tornado of 1840 traveled directly down the Mississippi, killing hundreds of people.
"The best place to be during a tornado is the southwest corner of your basement."
It's an old wive's tale that the southwest corner of the basement is the safest, derived from the idea that most tornadoes will approach from the southwest. The truth is, no one part of your basement is any safer than another. The most important thing to remember is to stay as far away from windows as possible. More on tornado safety here.
"If you're in your car on the road when a tornado is approaching, hiding under an overpass is your safest bet."
This is probably the worst tornado myth. Taking shelter under an overpass is one of the most dangerous things you can do when a tornado is approaching. The reason has to do with the way the tornado's winds could potentially interact with the bridge structure. At the very least, taking shelter under an overpass puts you at a higher elevation with no protection from debris and winds. The Norman, Oklahoma Weather Service office compiled a presentation on the topic after the May 3, 1999 tornado outbreak, which can be viewed here. More on tornado safety here.