Does chopping vegetables remove vitamins? I hear some chefs saying on TV: "Don't cut your vegetables in the salad too small, because the smaller you cut them the more vitamins are lost."

Is that true? 
and if it was really true, what is a way to make vegetables keep their vitamins while being cut into small pieces, becasue I like salads to be eaten when each vegetable slice is around a bite in size.
This phenomenon affects fruits much worse than vegetables actually. The FDA published a report that cut or peeled fruits will lose half their vitamin C content in 1-2 weeks. Over 10-25% of this loss will occur in fruits in only 5 days.

For vegetables there exists a similar, albeit less pronounced effect. When cut the flesh inside of the vegetable is exposed to oxygen and the protection provided by the peel or covering is lost. 

That being said, the process of these vegetables losing nutrients is not an immediate one. Realistically the loss that occurs between you cutting the vegetables and placing them in a salad is probably distinctly minimal. I wouldn't worry too much about the nutrient loss as long as you aren't cutting the vegetables and then storing them for a long period of time. 

See: http://nutrition.about.com/od/askyournutritionist/f/cutveg.htm