May I discuss a paper I have only seen as a referee? Recently I refereed a paper in pure mathematics. Although I recommended a "major revision", I was quite impressed with the paper, and indeed I would like to: (1) describe his results in a grant proposal I am writing, and (2) eventually apply the author's techniques in my own work.

The author has not yet made his paper publicly available (e.g. on the arXiv or on his personal website). Ethically speaking, may I now freely refer to his paper in my proposal, and later in my work? Or am I bound to wait until the paper is published or until the author has otherwise made it available to the public?

I e-mailed him recently, let him know that I was a referee for his paper, and asked his permission to do this. If he writes back to offer (or deny) me permission, then that settles the issue. But what if I do not hear back?
No.  It's a standard expectation that you'll keep confidential anything in the papers you referee.  Once the paper is published (or even made available online as a preprint) then you'd be free to refer to what has been made public.