This answer to applying sunscreen to dogs says that you shouldn't apply human sunscreen to them because it contains zinc oxide and that's bad for dogs. What about it is bad, and what would happen if a dog did come in contact with it?
The primary reason for concern would be zinc toxicosis. This is basically the formation of zinc salts in the stomach acid which are toxic and can lead to diarrhea, lethargy, seizures, and more. It's quite dangerous and can require surgical intervention to treat.
Common sources of zinc poisoning in dogs and cats include the Lincoln penny(!) and zinc-oxide creams. Dogs are more likely to get this problem because they more freely eat whats around them, but using human sunscreen on a dog or cat is very likely to result in them attempting to lick it off and, well, poison themselves.
Zinc oxide is SAFE for topical use. All the concerns regarding its potential toxicity refer to the ingestion of this substance. Dermal absorption of zinc from topical ointments based on zinc oxide is trace to none (study on humans). At worst, topical use could result in minor skin irritation. As long as it is not eaten, it is safe.
Zinc oxide on itself is also not inherently more dangerous to dogs than it is to humans.
A lot of chemical compounds are a lot more toxic to pets than they are to humans - for example, a common over-the-counter pain reliever, paracetamol (alias acetaminophen), is extremely toxic to cats; also, theobromine and caffeine are highly-toxic to both dogs and cats. However, zinc oxide is not such a compound, even though the related Q&A might potentially give the impression that it is.
Substances like paracetamol and caffeine are organic molecules that are metabolized by enzymes (mainly in our liver); cats and dogs either lack specific enzymes needed for metabolizing them, metabolize them much slower than humans and/or metabolize them via different pathways that lead to the formation of toxic by-products. On the other hand, zinc oxide is a simple compound and no fundamental differences between humans and dogs exist.
Actually, the reason for using dog-specific sunscreen is that sunscreens made specifically for pets contain a bittering agent to prevent licking and consuming; it is because a lot of sunscreen's constituents - not just zinc oxide - are generally not intended to be eaten and are not the most pleasant things to the stomach; also, I'd expect some of them to be worse in terms of toxicity than ZnO.
Zinc oxide is a relatively inert compound that is insoluble in water. However, it dissolves in most acids and alkalis; upon ingestion, it dissolves in stomach acid and releases free zinc ions. Zinc ion in solution is a strong Lewis acid and thus is corrosive - it irritates the mucosa of the digestive tract, causing stomach ache, nausea, diarrhea, and vomiting.
Upon ingestion, zinc ions are also absorbed and, depending on the dose, could cause zinc poisoning. Zinc is an essential micronutrient, which means it is required for life in small amounts - it is the dose that makes the poison. It makes zinc different than, for example, lead - which is highly toxic, is known to bioaccumulate, and has no biological role. In the case of lead, the only safe level of exposure is zero.
Also, inorganic forms of zinc (like zinc oxide) are poorly absorbed from GI tract. Dog food is often fortified with vitamins and minerals - including zinc. In the case of zinc, the most bioavailable forms are those in which the Zn atom is bound to a ligand (which is usually organic) via two or more separate coordinate bonds; this ligand is referred to as chelating agent or chelator and its bonds could be imagined as "claws" which are pinching/holding the metal atom - it stabilizes the metal atom and prevents it from reacting with the contents inside GI tract and being precipitated out in the form of insoluble salts, which would render it biologically unavailable. However, the cheaper brands of dog food are often using inorganic forms of zinc - like zinc oxide - and in this context, it is a problem because of how poorly it is absorbed from GI tract. I do not want to include direct links to the specific websites because of annoying popups that cause my web browser to malfunction, but to give some idea as to what to look for: I did a web search with the phrase "dog food zinc oxide" and two of the first three results were about zinc deficiency in dogs and how it could be caused by feeding dog food that has its zinc in a poorly-bioavailable form like ZnO. Of course, the ZnO concentration in dog food is going to be much less than in sunscreen - but it is just to give you an idea of its bioavailability from GI tract.
I attempted to find some articles and studies about dogs being poisoned as a result of zinc oxide-based sunscreens or other ointments, and the only ones I found did not involve the dog licking and swallowing the ointment applied on its skin, but rather the dog molesting the ointment container: squeezing a major part of its contents out and eating them, or chewing through the container and swallowing the contents together with fragments of the mauled container itself.
Nonetheless, it is still possible for excess amounts ingested to result in poisoning (but the same is true in the case of table salt). ZnO is odorless and tasteless, so it doesn't immediately produce repulsive sensations which would discourage the dog from further consumption.
LD50 for zinc oxide in this datasheet has two given values (from two different experiments): one of >5000 mg/kg (rat, oral) and the second of 15000 mg/kg (rat, oral).
For comparison, LD50 for table salt (sodium chloride) is 3000 mg/kg (rat, oral); for table sugar (sacharose): 29700 mg/kg (rat, oral).
Zinc competes with other micronutrients (like iron and copper) in terms of absorption; it means that zinc poisoning (especially chronic poisoning) could cause iron deficiency (which could result in anemia) and copper deficiency symptoms. If the poisoning did not result in death, the prognosis is good in most of the cases.