Is it safe to use plastic wrap in boiling water?

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.

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.

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

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.

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.

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.

Clingfilm is safe for using cold and wrapping cold foods to store in the (cold) fridge, NOT wrap foods up and put in hot water. This is insane. Please people, don't do this. Sorry to be five years late, but I was searching for the best ways to poach eggs and found this. This is NOT it.