Do wisdom teeth close gaps?

No: Teeth might drift slightly in some specific situations in a forward direction over time and perhaps close a nearby space, but it would not be directly caused by an erupting wisdom tooth. They certainly won't close generalized spaces in front teeth.

Movement: Don't count on 3rd molars or wisdom teeth to close gaps, diastemas, etc. Sure, there is always going to be movement of teeth, shifting, drifting, etc. In the mouth, but in order to close a gap/diastema permanently, you may need orthodontics with frenectomy, or may need veneers, or may need resin bonding only... All of these are better than relying on wisdom teeth to do it.

No: No they do not and they are not the reason why teeth crowd up after braces.

No: Wisdom teeth will not close a gap present in the anterior teeth. If a first (six year) molar is lost at a very early age a developing malpositioned wisdom tooth can cause some significant tilting of the adjacent second (twelve year) molar into the space previously occupied by the lost first molar. This is the primary reason why first molars should be maintained or replaced in the young patient.

No, but...: And the poor wisdom teeth get blamed for so many things. Spaces may close as the wisdom teeth erupt, but that's pretty much coincidental. If you have no wisdom teeth, those same spaces would close anyway. If you have gaps you should talk to both an Orthodontic Specialist about repositioning teeth, a General Dentist about reshaping teeth, and perhaps the best treatment is to do both.