Should designers base their font choices for body text and headlines on the typeface used in the logotype (if used) to achieve brand consistency?
Short answer: It depends.
Long Answer: If you need to enhance readability it is not a given. It all depends on your typographic contrast. Sometimes you need branding to be more important and if the typeface is readable enough to be used you an go for the brand typeface. In other cases you may want to take a typeface that has good contrast and high legibility.
So if it's a content heavy site, i'd go for readability and pick a suitable typeface (which is not necessarily brand, so the answer is really no ;) ).
Remember: content is king!
Typefaces chosen for logos are chosen based on different criteria than typefaces used for headlines. Typefaces used for headlines are chosen based on different criteria than typefaces used for body copy. Etc.
That said, all of the typeface choices should work together under the larger umbrella of the branding system, so yes, brand consistency is important, but consistency isn't necessarily achieved by using the same typeface for one task as another.
This really isn't a question for this site's format and I'm sure it will be closed soon enough, but in order to give you another perspective:
From a business standpoint, no. You never use your logo typeface in collateral, website, etc. Your logo is supposed to be unique and stand apart for easy recognition.
What happens when you use your logo typeface for website headings is a dilution of your brand. This decision has nothing to do with readability or nything else design related. It is purely business and research done in that arena for the last 50 years.
Now if you used Arial in your logo, you can safely use it in your body copy. However, while doing so you should also fire your logo designer for using Arial for your most distinctive mark. Conversely, if you use Comic Sans in your logo, you should also fire your designer and not use Comic Sans in your copy in order to keep your mark unique.
This blogpost is a blog post and should be taken as that, but it also illustrates this same point. These aren't experts by any means but good points:
"What I’d never recommend is using the font you’ve used in your logo in your body copy. Quite simply because it weakens the logo. By all means use a complimentary font, [...] or use a contrasting font. But try to avoid using the same font. It makes your logo look less of a logo and more of just type."