Do I have to ask a research advisor before sharing a technical report? After finishing my bachelor's degree, I received a fellowship via my home university to conduct independent research at a university of my choosing. All of my living expenses and stipend were taken care of by this fellowship, and the project I worked on was my proposal. I reached out to professor Z to see if he would host me and supervise the project if I were to receive the fellowship, and he said yes.

I'm now applying to PhD programs, one of which gives the option to submit a research writing sample. I would like to submit the final report I wrote up on my project, but I'm not sure if it requires my advisor's permission. This report would presumably be shared among the graduate admissions committee. 

Some other possibly relevant info: there are no plans at this time to publish the results. The project wrapped up over a year ago now. There is no ongoing research on this topic in professor Z's research group (which I know because I keep in contact with a few students in his group). The report does not (to the best of my knowledge) contain any information on ongoing projects within Z's research group.

The deadline for my application is soon and I haven't received a response from my advisor yet (which is my fault for not asking him sooner, but not relevant to the actual question I'm asking). If he doesn't respond, I've told myself that I won't submit it. I have actual published work that the committee can pull up if they like, I just thought it would be a plus to show the results of a research project that didn't lead to any publication, but which highlights my research abilities anyway.

I'm curious though what the etiquette is in this case. Would it be a major faux pas for me to submit this writing sample with my application?
I haven't understood why you were hesitating.  You shared several reasons why it would not be a problem.  You made a convincing case that there would be no danger of some publishable results being scooped.

I suppose now that you asked Prof. Z's permission, with brief lead time, it could be awkward if you submitted it and then he wrote back subsequently to say no.  You could get around this, however, by writing again, to ask if he received your email, and to say that you would very much like to submit the report in your application, but will omit it if he notifies you by (date).  Put something like "time sensitive" in the subject line.

If you're asking just to satisfy your curiosity, sort of in hindsight, then my answer would be no, you wouldn't need to ask permission in the situation you described.