Is it bad if a chameleon swallows a bee?

It is possible to get stung while swallowing a stinging insect, though a chameleon would usually crush the insect before or while swallowing. It is probably more likely to get stung around the tongue than the throat. This could just result in temporary discomfort, but a bee will normally detach the stinger and sac when stinging invertebrates, so a potentially harmful dose of venom can be administered even after the bee is swallowed or crushed.

A chameleon should be fine eating many types of 'bug'... but if there is obvious swelling around the throat, tongue, mouth or jaw, and/or the chameleon is having difficulty eating (or shows disinterest in food), you should consult a veterinarian. Be observant for swelling or worsening of appetite coupled with uncharacteristic lethargy. Simply consult a vet if you are concerned. Better to be safe than sorry.