What is the meaning of 要 in these sentences?
All three examples can be interpreted as a future auxiliary.


  他很忙，但是他「應該要」來參加我的生日晚會
  
  He is very busy, but he「should be」coming to my evening birthday party.


This「要」is used in the sense of will, going to, like 我明天要上學 (I will be going to school tomorrow). Similarly,


  要實現夢想不是一件容易的事
  
  To realise (one's) dreams is not an easy matter
  
  要把漢語學好不是一件容易的事
  
  To learn Chinese well is not an easy matter