How dangerous is it to refreeze meat that has been thawed?

In theory you could thaw and refreeze as many times as you like, though the changes in temperature would definitely alter the quality of the meat's taste and texture.

What matters most is how long the meat has been in the so-called "danger zone" speaking from a temperature perspective. The "danger zone" is defined as being between 41 to 135 °F (5 to 57 °C).

Here is an excerpt from The Professional Chef, by the Culinary Institute of America:

Foods left in the danger zone for a period longer than four hours are considered adulterated. Additionally, one should be fully aware that the four-hour period is cumulative, meaning that the meter starts running again every time the food enters the danger zone. Therefore, once the four-hour period has been exceeded, heating, cooling, or any other cooking method cannot recover foods.

As the water in the muscle fibers freezes it expands and creates a mushy texture. The reason that commercially frozen meat has less degradation of texture is due to the speed at which they can freeze things. The quicker that freezing takes place the smaller the ice crystals will be. Home freezers are best at keeping frozen foods frozen but take much longer than commercial freezers to do the actual freezing.

If the food was properly thawed, re-freezing once will probably have a minimal impact on texture but it's going to depend on the item. Ground meats such as ground beef probably won't be noticeable vs. a steak or other cut that normally has a fairly tight muscle structure.

In addition to freezing altering taste and textures by damaging cells and co, there are safety reasons to avoid refreezing. In France at least, all commercial frozen food have to display the "never refreeze unfrozen products" warning on the package.

The official explanation (I didn't look for scientific studies) is that when you defreeze on the first time, frost-proof bacterias will have few competitors because the initial freezing has killed most of other bacterias, they will have easy-to-digest food because the initial freezing has dismantled cells, and they will have further ideal development conditions because of the slow increase of temperature. So at first, it's likely to find more bacterias of a single kind in unfrozen food than in never frozen food with the same unfrozen lifetime (of course you may have more bacterias in the never frozen food, but of many species, I guess), though not at a dangerous level. Then, unless you have industrial equipment or your dish is packaged in very thin layers, the refrozing is slow, so bacterias have even more time to develop in such good conditions. So after being fully refrozen the product will have much more bacterias (of each surviving frost-proof species) than the original, possibly at a non-safe level. Of course it will be even worse next time, since bacterias follow an exponential growth (I guess once they're not in the exponential phase anymore, it's way too late already…).

So it seems not only the danger zone issue is hidden (one wrongly assumes that once in the freezer, the food is not in the danger zone anymore, but it takes some time to freeze, depending on the size), but there are specific issues because of conditions generated by the freezing.

On a side note, it's surprising how safety advices depend on the country (I guess). People on cooking.SE usually strongly advise to follow the US agencies "2 hours in danger zone" recommendation (interesting to note that the quote on JYelton's answer mentions 4 hours instead of 2). French (European ?) agencies recommend the same, but insist much more (it's my feeling at least) on the refreezing issue, and I'd bet much more French people are aware of the latter than of the former (probably because of the mandatory mention on packages).

Disclaimer : I'm no physician or food or health or food safety specialist, just reporting informations gathered on trustable (IMO) websites.

The answer depends on how the meat was thawed. If you read any of the health and safety documentation it tends to stipulate that meat thawed in a refrigerator can be safety refrozen. Meat thawed by other methods, particularly if the temperature reaches 40°F–140°F (4°C–60°C) should be cooked before refreezing.

It's not dangerous but it significantly impacts the quality of the meat. Most noticeably, it's ability to hold onto moisture.

Frozen food should be consumed quickly after it is defrosted. Do this within 1 week after the first defrost and 24 hours after the second. Red meat is the fastest decaying food and it's already been frozen before getting into your freezer.

A study indicates that refreezing food adds risks for contamination because freezing the meat ruptures the muscles in such a way that bacteria can travel from the surface to the interior. Each additional thaw allows the bacteria to expand in areas normally not affected by the bacteria.

http://news.bbc.co.uk/2/hi/health/2072755.stm

Supposedly flash freezing causes smaller ice crystals, which minimize this from happening.