Unless you have severe allergies or have an aggressive dog that bites you, I don't see how your dog could harm your health. Except maybe if you become so obsessed with your dog that you cease to go out and socialize with other humans, which could impact you emotional/mental health.
But overall, dogs do wonders for your health. They provide companionship which helps you from becoming lonely, they practically force you to exercise, it's proven that petting a dog can lower stress levels and blood pressure, for people who are disabled, having a service dog helps them to become more independent.
As for me personally, I'm painfully shy and suffer from bouts of bad clinical depression (stemming from a very bad case of Inattentive ADHD), both of which make me want to completely withdraw from the outside world; but having my Toy Fox Terrier, Penny Lane, forces me to go out into the world to give her walks, buy her food, take her to he vet, etc. I also have anxiety issues and she always seems to know when I'm anxious and will come sit next to me and put her head on my lap and I'll pet her and it calms me down A LOT. Or she'll go get a toy and pester me until I go play with her, which gets my mind off whatever I'm worrying about. Also, I tore the ligaments in my ankle back in July and was bed bound for a couple weeks and she stayed right next to me the entire time, which prevented me from getting lonely. I seriously don't know what I'd do without her... :)