Will these "errors" in my advisor's letter raise red flags when I apply for academic faculty positions? I am currently on the job market. One of the positions uses the online application system where I am able to open the letters of recommendation and read them. I found out that there are "errors" in my advisor's letter. 


First, she put the name of University A in the letter that goes out to University B. 
Second, she wrote that I have successfully defended this month when her letter was dated June, which I worry is going to give the search committees the impression that I defended my dissertation in June when in fact I had defended in May. 


I conveyed this to her; she offered to fix the name of the university. But what if the same error is repeated in her second letter, or worse, she adds more errors in her second letter? 

So my questions are: 


Will these errors raise any red flags in the eyes of the search committees? 
Will these errors give me enough reason to not to use my advisor's letter? 


I am contemplating not to list my advisor as one of my references. I have read the responses to a question similar to mine in this forum, so I understand that I am taking a risk and I will have to explain it if I choose not to list my advisor as one of my references.
If I were your advisor, I would fix these typos before sending out any more letters (i.e., to other schools).  If there is an online system where I can instantly and unobtrusively replace a letter with a fresh copy, then I would do it.  Otherwise I would let it go.

These are not serious enough for you to worry about.  The impression that this will make on the reader is that your advisor is a human being, perhaps a human being who is rather busy.  Ideally your advisor will be someone whose reputation is known to the readers, so cannot be spoiled by a few careless typos.


  One of the positions uses the online application system where I am able to open the letters of recommendation and read them.


This concerns me...but you didn't ask about it.

Added: I missed at first this part of your question:


  [W]ill these errors give me enough reason to not to use my advisor's letter? I am contemplating not to list my advisor as one of my references, and I have read the responses to a question similar to mine in this forum, so I understand I am taking a risk and I will have to explain it if I choose not to list my advisor as one of my references. 


Heck no.  Not getting a letter from your advisor upon graduation is a very bad idea in almost all cases.