Do Any Teeth Have to Be Removed Before Getting Braces?

In my opinion, teeth should only be removed when absolutely necessary to gain space to align the other teeth.  Whenever possible it is better to retain all the teeth and develop good broad arch form.  Many people don't realize the connection between well developed dental arches, airway (sleep apnea), and lower jaw position in relation to the base of the skull.  When teeth are removed unnecessarily the result can be less room for the tongue to posture, less open airway, more difficulty with good sleep, and a jaws that are retracted.  This can lead to imbalanced muscle activity and TMJ problems

Whether or not extractions are needed prior to braces depends on the severity of the crowding.  In any situation where extractions are recommended I would recommend getting another opinion from a dentist or orthodontist who practices functional orthodontics.  Many times, expansion can be done to develop the dental arches properly and make room for all of the teeth.  When dental arches are developed properly, this can result in the best facial balance and esthetics.  

With more traditional orthodontic treatment (braces only) many times permanent teeth have to be extracted when crowding and overbites are present.  Adding what is called Dentofacial Orthopedics (fancy retainers to expand bone structures) prior to braces can eliminate the need for permanent teeth extractions in almost all cases.  In my practice, permanent teeth have to be removed in well under 1% of the cases.  Many of the before and after pictures I submitted used Dentofacial Orthopedics and none needed permanent tooth removal.

This depends on the individual situation. However, in my practice, less than 3% require extractions vs. the national average of 15-20%. I prefer not to extract, when possible, so that natural dentition is preserved. 

This totally depends on each individual situation. If the teeth are too crowded to move without the extra space gained from extracted teeth, they should be removed prior to any tooth movement. If there is enough room to align them first, they can be removed during treatment to correct issues such as profile, midlines, open bites, etc. When trying to move upper teeth back (distalization), you may require the extraction of your wisdom teeth to create room for the other teeth to shift backwards. Note that you can have braces placed, without a wire, and no tooth movement will occur.
If you are having braces placed in an indirect bonding technique, it is vital to have the teeth extracted after the braces are placed. Otherwise, the teeth may begin to shift away from the position from the original position, and the custom trays made for your teeth may no longer fit properly, leading to improper bracket placement.
Extractions with Invisalign require proper timing to be done just before your initial aligner delivery, unless you wear something to retain your teeth in position after the extractions. Having teeth removed much earlier may again allow teeth to shift, making your initial aligners less conducive to fitting properly.

Modern materials and techniques allow us to avoid extracting teeth, while in the past it was routine to extract. While it cannot be avoided sometimes, if arch expansion is possible, it is usually advised.