不要说话 vs 不说话 (vs 别说话)
'要' in '不要' cannot be removed, because '不要' is a compound word that mean 'don't'(auxiliary verb) which is different from '不' (adv: not)


不要说话/ 别说话 = Don't talk (demand) 
不说话 = 'not speak'
看电影的时候请(不要)说话 - Please (don't) talk when watching movies (O)
看电影的时候请(别)说话 - Please (don't) talk when watching movies (O)
看电影的时候请(不)说话 - Please (not) talk when watching movies (X)


他(不说话) = he (doesn't speak)