The damage in the close up photo is secondary. That is, it occurred after the branch was already dead. The bark is decaying (maybe being digested by insect activity) and it appears to have some borer larvae exit holes. I can't be sure what made this branch die in the first place, but it likely is not "contagious" at this moment in time. Remove it completely and toss it out. Although not contagious, if there are disease issues going on, it may house spores that would be a potential problem later one.
One possible reason for the original damage is fire blight. Here are some links to more information on fire blight, so you can watch for it.

http://www.ipm.ucdavis.edu/PMG/PESTNOTES/pn7414.html 

http://extension.arizona.edu/sites/extension.arizona.edu/files/pubs/az1030.pdf

Thanks for your response.   We removed and burned the dead branches, and all that was left looked green and healthy.  Now, a few weeks later, some of the top branches are doing the same thing and dying rather quickly.  Some things do suggest fire blight, for example the fruit on that branch has blackened.  But I don't notice anything oozing or any pink or orange streaks.  There are a few grey spots on the underside of leaves from otherwise healthy looking sections.