Not enough recommendation letters
If I submit my application with only two letters, will it be rejected as incomplete?


Yes, probably.  That is what the word "required" means.  It is possible that some employers might consider the application anyway, or perhaps give you more time to submit a third one, but many will not.

They want to make sure they are getting a range of opinions on the candidate's qualifications, to reduce the chance that they are just hearing from one or two biased people writing inaccurate letters.


  Is it a bad sign if I send only two letters in those cases where are requested at least three or more?


Yes.

So you should try harder to find a third person to write a letter.  We have lots of questions on this site about how to find letter writers.  For instance, you can tell your other two writers about the situation, and ask if they have any ideas.