Ask Mr. Smarty Plants

Mr. Smarty Plants couldn't find any reference to Pyrus calleryana 'Bradford' (Bradford pear), or any Pyrus spp. for that matter, in most of our favorite poisonous plant databases:

Poisonous Plants of North Carolina

Cornell University Plants Poisonous to Livestock

Canadian Poisonous Plants Information System

Universiy of Pennsylvania Poisonous Plants

Texas Toxic Plant Database

However, one database, Toxic Plants, from University of California-Davis lists the seeds of Pyrus sp. as "Major Toxicity:  These plants may cause serious illness or death. If ingested, immediately call the Poison Control Center or your doctor."  American Dog Trainers Network also lists pear seeds on their Poisonous Substances list.  The Humane Society of the United States does not list pears or pear seeds in their list of Common Poisonous Plants, but they do list apple seeds. You can look for more reports of the toxicity of pears/pear seeds by Googling "dogs poisonous plants pears".

WrongDiagnois.com had this to say about poisoning by seeds of Pyrus communis (common pear):  "Pear seeds that are chewed to a very fine consistency or crushed first may result in the release of glycosides which can be turned into cyanide by stomach acids. Large quantities are required so the risk of even mild poisoning is very minute." 

Of course, you don't want potentially poisonous substances about for your dog(s) to eat, but you probably like your pear tree as well.  If this were a common pear with large fruits, you probably could get by just watching for the fruits to drop and removing them from the area.  However, since these fruits are so small, I would think it would be harder to pick up all fallen fruits and easier for a dog to crush the seeds if it decided to eat the fruits. Since the Bradford pear is non-native and considered invasive in some parts of North America, perhaps you could consider replacing it with a native ornamental flowering tree such as:

Cercis canadensis var. texensis (Texas redbud),

Cornus florida (flowering dogwood),

Cornus drummondii (roughleaf dogwood)

The sap of Cornus spp. is reported by Toxic Plants UC-Davis to possibly cause skin irritation. Cornus spp. weren't on any of the general or dog-specific sites listed above for dermatitis or any other toxicity and coming in contact with the sap is not too likely to occur with dogs.

Before you pick a new tree to replace your Bradford pear, check the toxic plant databases above before you buy.

Cercis canadensis var. texensis

Cornus florida

Cornus drummondii