How do bumblebees and hornets avoid the negative effects of inbreeding?
Queens do not generally breed with their brothers, but with males from other nests. In the life cycle of bees (and other social Hymenoptera), new queens are born late in the season along with haploid male drones. These all leave the nest and disperse in the landscape to find mates to reproduce with. After mating, all males die and the queens overwinter to start new colonies next spring. The situation is therefore not all that different compared to other annual, semelparous (single reproductive event) species, since you have population mixing before mating and reproduction. 

Here are a couple of mechanisms that bees avoid inbreeding:


Population mixing before reproduction (see above).
Inbreeding avoidance
That queens prefer not to mate with close relatives has been shown in several species, see e.g. Foster (1992) and Whitehorn et al (2009).
Polyandry
Each queen generally mates with several males (regulary more than 10 in honey bees, see Palmer &amp; Oldroyd, 2000) and stores their sperm to fertilize eggs during the next year, which means that workers and the queens of next year will have different fathers (increasing genetic variability).


References:


Foster. 1992. Nestmate Recognition as an Inbreeding Avoidance Mechanism in Bumble Bees (Hymenoptera: Apidae). Journal of the Kansas Entomological Society 65(3)  
Whitehorn et al. 2009. Kin recognition and inbreeding reluctance in bumblebees. Apidologie, 40(6)
Palmer &amp; Oldroyd. 2000. Evolution of multiple mating in the genus Apis. Apidologie 31(2)