Ok, twice now (the second time was today) I have made the bonehead mistake of forgetting to refill my propane tank before brewing and running out of propane mid-boil. It takes me 30 minutes to go get more propane and get the wort boiling again. I have just acted like no time elapsed between the flameout and the next boil (so if I am 30 min into the boil when the flame dies, I pretend like I am still 30 min into the boil when I get the boil going again. Thusfar this has only happened after my 60 min addition and before any other additions. What exactly is this doing to my beer? I mean - I didn’t notice anything odd with the previous beer, but I am curious what’s happening and what kind of problems this might bring.
That where you get your propane might not be open next time? 
I always “give a lift” to the tank to see how much I have left. Although I usually have full backup tanks at home or I can steal from the grille if need be. Plan ahead or buy a second tank and have it as backup. I doubt your beer is affected by what you have described, especially not adding your later addition hops until the wort comes back to a boil.
I once had to interrupt a boil similarly, and the beer turned out fine. I didn’t make any adjustments to account for the interruption.
You may already know how to tell how much propane you have left, but just in case you don’t: the tare (empty) weight of the propane tank should be stamped on the side – usually around 17 lbs. Step on a bathroom scale while holding the tank, subtract the tare weight from the reading on the scale, and you’ll get the weight of the propane left.
I definitely recommend getting a second propane tank. I have run out of propane during a brewing session but fortunately it was while heating the strike water. As far as how it will affect your beer… it could cause your beer to be more bitter by extracting more hop alpha acids since the wort will still be hot enough. Fortunately, you did not add the later hop additions yet. Other than that I’m sure your beer will turn out fine. 
I have three propane tanks. Also, check into an immersible heating element. They save a lot of propane.
FTR at my brewery I run all electric (my kettle is a 55 gallon blichman with two low density, high wattage elements.) I would love to have some kind of electric rig at home and if I ever make any money I might set myself up an all electric brewery out on the back deck.
Why brew at home anymore? Get yourself a “pilot brewery” in your nano brewery. It’s a tax writeoff, as well as the utilities, etc.,
All my homebrew stuff is down at the brewery and is my “pilot” system. The 55 gallon Blichmann system is going to be my pilot system soon as we are about ready to upgrade to a Premier Stainless turn key.
As far as “why brew at home” … just because I think I would still enjoy homebrewing at some point. Don’t have time to do it right now though.
I’ve ran out once near the end, we now have 6 tanks with 4 full at any given time.
< edit to fix FAT Finger typing >
One of these days that will happen to me too. I always forget getting the propane tanks filled.
As to what happens, here is what I think. Isomeration and protein denaturation continue while the wort is hot. Maybe a bit slower if the wort cools down significantly. The same goes for the creation of DMS.
What I suggest is that you assume the idle time as actual boil time. Regardless of that you should also make sure that you boil the wort at least 10-15 min at the end. Possibly with a higher boil-off rate than usual. This will drive off DMS and other unwanted volatiles.
If possible keep good notes and, if you want to give us more data, you may also brew that batch again but then w/o running out of propane.
Kai
I have three tanks. As soon as one becomes empty I exchange it. I would like to go electric some day. I understand it to be cheaper than gas.
I had an emergency once that made me quit for a while during a boil. I noted how long it had boiled, got it to boiling again and completed it with the rest of the required time. I couldn’t tell any difference in the final beer and it was one of my house standards.
One thing I’m liking about new technology are these automated propane lockers. We have about 5 within a 10 minute drive. Sure, you might not get the best deal, but I’m usually on a tight schedule and the chatty hardware store guys get on my nerves. I, too, have three tanks in cycle. One’s usually hooked up to the smoker, but since it doesn’t burn much, it’s a reliable backup.
As for affects on your beer, you’ll need to run some tests, I think. You might find a new brewing process. Heat-bursting?
When I run out, I usually add another 5 or 10 minutes to my timer. I figure that it stopped boiling before I noticed it being out or it at least wasn’t at the full boil.
I think this pointed me toward a replication problem I had a year ago. It was the day after Christmas and besides brewing in a <10F snowstorm, I’m pretty sure I ran out of propane mid boil and spent 15 or 20 minutes digging another tank out. I wasn’t able to get the same recipe to come anywhere near the profile I had for that brew. Maybe I need to try a split boil to see what it does.
Three tanks here as well, although one of them is attached to the grill but could be scavenged if necessary. Once I kick one, I whistle up these guys:
Then click, click, click, and they come deliver a new tank. They usually send at least one $2 off coupon a month via email and I usually wait for that to show up unless I’m really low. It ends up being about the same price as taking it someplace and exchanging it. And I don’t have to lug a propane tank around. I just leave it on my doorstep in the morning and have a full tank when I get home at night.
This happens to me fairly frequently, luckily there is a gas station with propane bottles a couple blocks away so I am back up and running within 10 minutes. I watch the freeze line on the side to make sure I don’t get stuck at the very end of a boil where it would be more critical.
Just playing with a recipe on Beersmith IBU contribution (Tinseth formula) of 1 oz 9.5% Perle are:
60 min: 32.3 IBU
75 min: 33.7 IBU
90 min: 34.5 IBU
I’m not at a point where I am making less than 10% adjustments to recipes anyways.
I was all electric with hot water heater elements until I started having problems with it. It was a cobbled together system and I never fully trusted it.
Leary of just the problems in all the prior posts I went with natural gas fired burners. Never run out in the middle of the boil. I’m sure the BTU’s generated are much less. I can tell when it’s been a brew intense month when I look at the bill!
“I just leave it on my doorstep in the morning and have a full tank when I get home at night.”
Wow! and to think I doubted the Propane Fairy! 
I have 4 tanks, including the one on the grill. I get them filled at the RV place. Last time I got 3 filled for $33 bucks. Much cheaper than the “exchange” type deals.
i have natural gas from the house to the grill with a connection set up for a natural gas burner (not yet purchased) however for you liquid propane users, i was at the home away from home depot last night and saw that they have 100 pound tanks available. or you could just get the big ass tank like mom has on the farm… ;D
I just went natural gas, but have been there with the “running out of propane” crew. I’ll echo that if you can find a propane business, it’s a lot cheaper to get your tanks filled there than doing the blue rhino swap at the 7/11.
If I’m doing just a five gallon batch I can put it in my 30qt pot and boil it in the kitchen. It happened once. The beer was fine. I use a 40 lb tank and have a 20 lb back up.
I heat most my water on the stove. I brew just out the back door on the porch. Packing hot water is not a big deal to me.
I bet I could fit three big burners like my stove top uses under my 60 qt pot if I built up a brew stand. Seems easier to me than hot water heater elements. One of them has no problem with five gallons. I pay less than 8 cents a kw/h here. It might be worth doing someday.