Clearly you lose mass. That’s not the question. I suspect that the flatus consists of CO2 and CH4. The latter is lighter than air and the former heavier, but I suspect that for people, the CO2 predominates by a lot (I don’t know about cows). So which is it?
I admit I don’t know, but foresee a hellacious diet book.
“Fart Your Way To A Slimmer You: The All-Bean Diet”
Assuming farts have the same density as the air we breathe, I’m gonna go with “neither.”
This is my reasoning: You could cup your hands around emptiness (that is, air) and imagine that the chamber you’ve created is airtight. Now, squeeze your palms together and let the trapped air escape. Would this have any impact on your weight? No.
Mythbusters actually analyzed the contents of a flatus once. I’ll see if I can track down their findings.
According to this cite/site, flatulence is almost all nitrogen, oxygen, carbon dioxide, hydrogen, and methane. The amount of methane is affected by genetics, while the amount of hydrogen is affected by colonic flora*. It doesn’t really get into percentage ranges though that I can see.
*band name
That’s why I was hoping to find the Mythbusters data – if I remember correctly they did get into percentages for their experiment. Unfortunately, while I can find a lot of sites that state they found the myth “Can you suffocate on your own flatulence?” was busted, none of them go into any details of their test.
This is a great conversation starter for the dinner table.
Plus we have beans on the menu tonight.
Should be a brisk discussion.
I’ll let you know what my geniuses come up with.
What they come up with at the dinner table…or later on? 
I too didn’t find a detailed fart summary from Mythbusters, but this article says:
59% nitrogen, 21% hydrogen, 9% carbon dioxide, 7% methane and 4% oxygen—all essentially odorless. (Less than 1% is what makes farts stink.)
It’s gas under pressure (relative to the outside air), so it’ll be denser than the same gases once they’re released. For that reason, I’ll guess that you lose weight during the event.
Didn’t Cecil write a column on this years ago? I’ll track it down later if I have time (and if it exists).
It must be that you lose weight. First of all, as the OP says, you lose mass, but I doubt you become any less buoyant – that is, you don’t float more in the air due to being denser, to offset the mass lost. As @Robot_Arm mentions, that gas is under pressure.
When you fill up a balloon with air, it weighs more than an empty balloon.
The balloon is the exact analogy as it depends on what gas it is filled up with. Helium filled balloons float remember.
That calculation is not complex even if I’m too lazy to do it, assuming those percentages are correct. I highly doubt intestinal gases are under enough pressure to significantly change density. Given how much more hydrogen and methane than room air I’d guess less … except even a few … um … bits of particulate material … in there easily fill the scale the other way. And I am quite sure most farts have some aerosolized solids.
Lose weight.
Ew. 
And even if you don’t want to think about that, there is a higher water vapor content of any gases in the body than in room air, which also adds significantly to density relative to room air.
A factor to consider in the other direction though is temperature? Internal gas is often warmer than room air so less dense.
I assume that, if you’ve been drinking Fizzy Lifting Drinks, farting makes you gain weight.
In order to establish whether the release of gas results in an increase or reduction of your measured weight, you would need to consider both the change in mass (i.e. the mass of gas released) and the change in volume of your body.
Buoyancy is relevant here to what we mean by “lose weight”. Even dense objects such as iron nails will weigh slightly less if weighed in air than in vacuum due to the weight of the air they displace (the difference is slight but is real and measurable, and must be allowed for when carrying out very precise measurements).
I doubt very much if the gases inside your body are pressurised to any great extent. The sphincters that retain the contents of your alimentary canal would not be able to withstand any significant pressure difference. I haven’t been able to find any numbers though.
Cecil’s take: You may weigh an infinitesimal bit more (although his answer wasn’t as definitive as I remembered it being).
Normal air, assuming 78% nitrogen and 22% oxygen, will have an average molecular mass of around 28.9. That mixture of gases (assuming the percentages are by moles, which would make sense for gases) will have an average molecular mass of around 22.2. The flautus gases will be pressurized some inside the body, but to pressurize them enough to get the same density, you’d need to be able to sustain a pressure difference of 0.3 atmospheres, which seems highly implausible. So flautus gases in your gut will, in fact, be buoyant.
Oh, and to expand on that: If those proportions are actually by mass, that means a larger number of the smaller molecules, and a smaller number of larger molecules, which would mean an even lower overall density. And I was also assuming an equal temperature for internal flautus and the ambient air, but the interior temperature is almost certainly going to be higher. There’s also water vapor to consider, but that’s an even lighter gas than the average of the given gases (molecular mass of 18), so assuming a given pressure (air pressure plus whatever the maximum pressure difference is the gut can maintain), that’ll make our flautus even more buoyant. So to the extent that my figure is off, the answer will certainly still be buoyancy.