How do I deal with an old reference letter from an advisor for job applications? I wrote my graduate advisor asking him to updated his reference letters for me, but he has yet to respond. I e-mailed him again, but he and I did not have the best relationship, and I am worried that now that I have graduated, he may not do this for me. I was curious about I should do in the situation that he refuses to update it.

I have been told it raises questions when a letter is not included in applications.

His old letter is stored on mathjobs.org, and the last date was probably April of this year. I was curious that, if I have to, would it look bad if I use the old letter?

If anything, since I have deadlines coming up, can I use the old letter as a placement? I was told I can mention that a letter should be updated soon in my cover letter.
Not exactly an answer, but a couple things to note.

Letters on MathJobs expire eventually; I think you can see the expiration date on the "cover sheet" page. It will be very awkward if it expires partway through the application process. 

Also, note that your advisor will be able to remove or update the letter at any time, which could be a problem if he is really vindictive - he could replace the letter with something unfavorable. 

So in short, regardless of how an old letter looks to the committee, it may not be wise to use the letter against your advisor's wishes.