I think it depends on the dog and your training, of course. I had a fox terrier and beagle mix that I loved to pieces (RIP Sadie), but I would never let the chickens out when she was in the yard. Her terrier instinct to kill small animals was just too strong. She also would bark at the chickens and try to challenge them through the run fencing when I gave them a treat that she really wanted (like stale bread). My other mutt dog (have no idea of the breed mix) is very trustworthy around the chickens. I do get concerned that she could accidentally run over one as she charges around the yard when another dog is walked by, but so far so good.
I would definitely be wary of any terriers. They've been bred for hundreds of years for the express purpose of catching and killing vermin. If one of the chickens decides to dash around (like they sometimes do, especially when young), the terrier instinct may just kick in and the dog could grab the bird. Even if the dog lets go of the bird quickly on command or the thinking part of its brain kicks in and it realizes that it's not supposed to do this, the chicken could be injured.