Agree with ceejayoz that melting isn't the issue.
Melting is just the upper bound for problems.
The common concern is about leaching , the diffusion of chemical toxins from the plastic into the food.
It's known to be accelerated through fats and at heat, hence the concern.
There seems to be consensus that "some" leaching occurs; the issue is that it's unclear how much, from which plastics, whether the body absorbs or excretes it, and if it matters.
Alarmists will tell you that the chemical run-off is toxic and causes everything from cancer to birth-defects.
Governments have looked into the issue but not come out with a hard no.
Industry groups are fighting the issue, but losing public opinion on things like waterbottles (see nalgene, which removed Bisphenol A (BP-
A)from their formula).
Chefs generally care more about results than fringe chances of non-lethal toxicity and they have been using plastic wrap for as long as it's been around.
What should you do?
It depends.
Weigh fear with convenience.
Use good quality products if you do use it.
Don't do it too often.
Or just enjoy it.
cooking.stackexchange.com/users/1443/Ocaasi
Official:
No definitive answer, there are different governing bodies and plastic wraps (cling films) in the US/Europe.
There are also different plastic wraps for different purposes, microwave safe, safe for use with fatty foods, so on.
Whether its safe or not will depend on the product used and most likely the time boiled for.
The UKs Food Standard Agency said no (in 2012!)
broken link - "Do not use cling films where they may melt into the food, such as in conventional ovens or with pots and pans on cooker hobs".
The cling film industry are understandably much more positive.
Practical:
I've poached eggs using cling film for ages and I'm still here.
Read the plastic wrap boxes for warnings and choose the plastic wrap that's most suited.
Poach an egg as described here
(as of 2016, link requires registration to read) and check if the wrap comes away cleanly, If so you're good.
I'd revisit this advice
if your recipe called for cooking like this for longer than 5-10 minutes, there are likely more suitable products/recipes for extensive boil-in-the-bag manoeuvres.
EDIT
Many meals have passed, I'm less gung-ho on heating cling film/plastics.
It's amusingly still as contentious as ever and I certainly wouldn't brush the film with butter/oil/fat.
My totally unscientific advice now would be learn how to poach an egg properly, It's fun, tasty, and significantly less boring than reading this answer.
I like Ocaasi's answer, FWIW.
cooking.stackexchange.com/users/746/paprika
This article (1) says that plastic can release cancerous toxins, while this one (2) says this is a hoax.
I would be careful and do some more research before using the plastic in this situation.
(1):
www.foodproductiondaily.com/Packaging/Research-suggests-cancer-link-to-plastic-packaging (2):
www.hoax-slayer.com/plastic-cancer-link-hoax.html
cooking.stackexchange.com/users/757/Gustavo Eulalio
Water boils at 100c(212f).
Good quality plastic wrap has a melting point between 120 and 140c(250 to 290f)
So the answer is, it's ok
, Just make sure you read the carton when buying the wrap, it should tell you the safe temperatures.
You can view my comments here
How should I poach an egg?
regarding the poaching of eggs in plastic wrap.
It's something I do quite often.
cooking.stackexchange.com/users/115/Pulse
The UK Food Standards Agency says "Do not use cling films where they may melt into the food, such as in conventional ovens or with pots and pans on cooker hobs."
Here is the link , this is the first bullet point under the second Q&A list of answers.
This was from 2002, 13 years ago.
cooking.stackexchange.com/users/39214/Lizzie F
Any non food grade plastic (and even those) when heated to certain temperatures (unless through incineration, around 1000c) will leach toxins, specifically dibenzofurans/dioxins https://en.wikipedia.org/wiki/Dioxins_and_dioxin-like_compounds
So no, using cling wrap under any circumstance other than cold, is not recommended, period.
cooking.stackexchange.com/users/49638/tito
While I've come across one or two recipes that call for using plastic wrap in pots of boiling water (for example, to  cook eggs ), I've always been a bit wary of putting the wrap in boiling water. Can someone confirm or deny whether it is safe (or a good idea) to put plastic wrap in boiling water?
Is it safe to use plastic wrap in boiling water?
cooking.stackexchange.com
<food-safety><boiling>
cooking.stackexchange.com/questions/1717
cooking.stackexchange.com/users/748/Blakomen
Especially with non-food grade plastics, high temperatures may cause toxins to leach into the water. Since products and government standards vary, choose a good quality cling film and check the carton for safe temperatures. Be aware that some plastics may leach chemicals into the water. Only use good quality products and read the packet before use.
Non-food grade plastics may leach chemicals into the water. Make sure you read the packet for safe temperatures. 