There are around 200 viruses that cause symptoms that we humans refer to as "having a cold." When we're sick, we don't really care which one did us wrong. Have you ever gone to the doctor for a cold to be told that rest, fluids, and TLC was all they could prescribe? In due time, viruses will run their course and you'll feel good as new. (But we hope you at least got a doctor's note for a day of rest.)
Most (but not all) viruses are species-specific. That means the chance of you - or any other human - passing a virus to your rabbit isn't worth stressing over. There isn't a viral equivalent in the rabbit universe to the group of viruses that causes most of our colds.
That being said, viruses aren't the only bad guys at play every time we feel under the weather.
Can rabbits get a cold?
Rabbits don't get colds in the same sense that we do. Pasteurella multocida is a bacterial organism that likes to take over in eyes, ears and nose, and can even cause abscesses. Gunky noses, crusty eyes, sneezing, and labored breathing in rabbits is usually a result of a bacterial infection. This is an important distinction. While it means rabbits are mostly safe from falling victim to our viruses, it also means their upper respiratory infections aren't going to get better with TLC alone.
A rabbit showing symptoms of a respiratory illness needs to see an exotic veterinarian for appropriate treatment. Upper respiratory infections can be caused by different types of harmful bacteria. Unlike the viruses that cause most human colds, these bacteria are more likely to get worse than better if left to their own devices in a rabbit's body. A simple URI can go south quickly and turn into something more serious, like pneumonia. Your vet will probably prescribe a rabbit-safe antibiotic for at least ten days to knock out the bad bacteria and keep it from returning. A doctor's note for a few days of rest won't hurt either.
What about the flu?
More good news here. The groups of viruses that cause human influenza don't translate to any bunny boogers. Multi-species households, however, take note: Ferrets are an exception! Ferrets and humans can pass the flu virus back and forth.
"Snuffles" is the catch-all term to describe the rabbit version of the human cold or flu. Again, these symptoms are probably a result of bacteria. The types of bacteria most commonly cultured from rabbit nasal discharge are Pseudomonas aeruginosa,Bordetella bronchiseptica, and Staphylococcus aureus. If no lab tests don't show one of these (or other) bacteria, however, it doesn't mean your rabbit has caught your flu or cold virus. A foreign body or dental problem can first present as a chronic runny nose or eyes.
So, can rabbits Become sick from humans?
Rabbits are highly susceptible to herpes simplex virus (HSV), which causes cold sores in humans. Aside from this though, current research shows the chances of transmitting disease from humans to rabbits is minimal to non-existent. Viruses look for certain receptor molecules to attach to like a lock and key; rabbits don't have the right receptors to be impacted by the viruses that cause our common colds (lucky ducks ... or buns).
The truth is, though, there isn't conclusive research that says human bacteria definitely ... 100% ... without a doubt ... CAN'T make a rabbit sick. Can rabbits not catch human illnesses, or do we not yet recognize how it presents in rabbits or think to look for these connections? Until the relationship between how bacteria affects humans and rabbits is more conclusive, it won't hurt to keep your distance until you've been on antibiotics for 24 hours and start to feel better. Hand washing before and after handling your rabbit is always a good idea.
In the vast majority of cases, there is no need to worry you'll cause your rabbit to become ill when you're sick. Rest easy and binky on.
References and Related Reading:
http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0089055
https://vcahospitals.com/know-your-pet/rabbits-infectious-disease
https://www.vetstream.com/pennardvets/html/Factsheets/Otherpets/24_286507.asp