I am working with a logo that is using ChunkFive Roman. What's the standard web design practices regarding this? Is it good design to use the same font as my logo for primary headers?
I don't think it needs to be the same as your logo, particularly if it's somewhat exotic, but it should complement your logo. If your logo has big chunky letters, the headline font should have something in the same genre rather than Bodoni. The logo should look like it belongs to the website, not that it had run there when it was a young logo, playing at hide-and-seek with other logos, and forgotten the way out again.
Your headline font should fit in with the font used in the logo, this is a good design practice. "Fitting In" could either mean mirroring the same look, or providing a suitable contrasting font partner instead.
ChunkFive has no WebFont variation, but the google font Alfa Slab is similar in that it is a Slab Serif as well as having a very heavy weight.
From this link you should be able to view suggested font pairings as well.
I would say that sometimes it is okay to have your logo font and header font the same, but not typically. The company Fjällräven does this and I really enjoy their design.
It does detract from the uniqueness of the logo font a bit, but in some cases works.