Forgive my ignorance, but what shrub or tree is this?

This is a white lilac

I could not find any lilac disease that is consistent with the browned flowers you are concerned about.  Bacterial wilt causes blackening of flowers, leaves and stems while powdery mildew infects leaves later in the year.  That said, I will note that the plant does not display good health and vigor.

I consulted with Ann Hazelrigg, the UVM Extension plant pathologist, and she agreed with my assessment.  I've included her comments, below.

Ann Hazelrigg - My guess is that frost damage caused the death of those early buds. it looks like the shrub is ok, because the next leaves/flowers produced look fine. THat being said, i think you are right, the shrub looks generally unthrifty and may not thrive if it is not in full sun. Looks like it is planted pretty close to the building too. They should make sure it is mulched, prune out dead portions and they may want to fertilize lightly with a general fertilizer. Water through periods of drought, and consider moving if the shrub is not in an appropriate place.