No, it will affect your TOD. I'm not sure sure how bad, but it will kick off and on. If all you have is a donut and the flat is on the back, take the front tire off and put it on the back and the donut on the front. This will minamize the on/off of the TOD a little bit.

I carry a spare when I'm traveling out of town, but other than that I don't carry a spare. I've been lucky to have one flat in the 25+ years I've been driving. For a little piece of mind, carry a can of Fix-A-Flat and tire plugs.
Last year in Moab, Tone got a flat on his side wall out on the trail. He didn't have a spare, but someone (I think it was Todd Addams) lined up 4-5 plugs and Tony was able to drive out of the trail.

Peace.
Tom