The Bumble bee (Bombus terrestris ) doesn't die after stinging because the "stinger" is constructed quite differently than that of the common honey bee. As a matter of fact, unless the victim of a bee sting is a mammal or other skinned creature (reptiles comes to mind), the bee is likely not to die after stinging. This is especially true when the bee is protecting its hive from other insects. The barbed stinger can inject venom into the intruder and be withdrawn without fatal damage to the protecting bee. Having been a bee-keeper for a number of years here in the western U.S., I've noted that neither species can bite. The mouth parts are designed for one purpose; that of acquiring nectar. They do have quite a long "tongue" for just that purpose.
The bees feet do have small barbs that could feel, to the barehand, like an attempt at biting, I suppose...
There are about 5 species of bee that do not sting, mainly in Australia and South America. We have one variety of bee here in the U.S. that's grown to aid in crop production that being the Leaf Cutter Bee. This bee does sting, but the venom is so mild it's hardly noticeable.