There are a lot of replies to this question and there are bits of an answer spread all over, but frustratingly the discussion has been dominated by CO2 emissions, rather than addressing the question asked. I'm going to collect the valuable information here and add in some of my own research. Please feel free to improve this answer, it is a community wiki (but you don't need to add any more about CO2).
#CO2 First and foremost, yes, the largest environmental concern in taking a flight is the CO2 emissions. Using the link provided by Rsf, a 10 hour flight could produce in the order of 900kg of CO2, a significant chunk of the average yearly carbon footprint of a single person. But for the sake of answering the question, we're only going to look at in-flight waste. Perhaps a large CO2 offset has been purchased to try and mitigate emissions, or perhaps an individual is tracking a CO2 budget and has accounted for the flight, but still wants to minimise waste. Either way, in-flight waste is what will be addressed.
#Waste According to IATA (thanks Zach Lipton for the link) a typical passenger on average creates 1.43kg of waste, of which 40% is food and plastic waste (~0.6kg). This is averaged across all flights, so presumably long haul flights will have more waste, and potentially a larger proportion of that will be food and plastic waste. So where does this waste come from, and what can we do?
#Snacks and miscellaneous For small service items, like drinks, peanuts etc. the simple option is to refuse. Peanuts will keep, and so will the cups, cans, bottles etc. etc. to be passed onto the next person. (To counter the anticipated "but it's going to be used anyway", the overall demand will be reduced, hence the airline won't purchase as much). For other things that might come in plastic, again you can just refuse to use them and leave them as they are. Be aware though that sometimes blankets can be wrapped in plastic, so if you think you'll need one, you might have to wear an extra layer just in case.
#In-flight meals For things like the in-flight meals, it becomes a bit trickier. If you haven't planned ahead, then the moment the tray lands on your table is probably too late to do much about it. The link provided by Dmitry Grigoryev suggests that in some countries, anything you touch on international flights will be incinerated for biosecurity reasons (although there is a push to temper these laws). However, if you aren't landing in one of these countries, or your flight is domestic, some of your plastic might be reused. To quote this news report from 2010: "Air New Zealand says its plastic utensils, including cutlery, trays, bowls, cups and plates, are sorted by staff for cleaning and reuse following international flights", although it doesn't specify if this is light plastic or not. It also mentions that plastic cutlery on some airlines is actually being sterilised and reused upto 10 times, and potentially as many as 30 times. Using it and returning it to the staff could be a reasonable option if you know the airline does this.
If you are unaware of how the airline handles used plastic cutlery, or you know they definitely throw it out, then the moment you receive the tray you should assume everything goes to waste. In fact, if you don't say anything before your flight, then there is probably already a meal set aside for you, on a tray, ready to go.
#Plan ahead
As noted by averell, budget airlines are often forgoing the traditional in-flight service, in favour of a "buy in advance" model (to coin a phrase). Choosing these airlines will allow you to virtually eliminate your waste by eating a sufficient meal before boarding, or preparing a light meal to take on board. If you are flying a full cost airline, then you usually have an option to specify your meal (people with dietary requirements will be well aware of this). In some cases you can request "no meal" and do the same as above.