Bad For Accessibility
Screen readers and other assistive technologies are often thrown off by dynamic DOM changes. They process & read the page in a linear fashion, and changing the content of the page after it's loaded might not be correctly handled.
There may be techniques to get around this, but I haven't looked into it too thoroughly.
Maintaining this kind of site could be tricky. For one example: If you created a new layout and changed the ID of the content area you were replacing with your AJAX links, it could break your navigation scheme in a pretty perplexing way.
This kind of AJAX behavior would also complicate any traffic analysis you may be doing; Google Analytics wouldn't properly register these AJAX loads without a manual call to
Adding more complexity to how your page operates also raises the bar for new developers; anyone working on the site would probably have to be made aware of how this behavior affects page loads.
Possible: Slower Page Load on Initial Visit
Each subsequent link clicked would be faster, but fetching the first page a user visited would actually take longer than a static version.
The reason I labeled this as a possible issue is that you could always send the first page statically (since you'll have the static version as a fallback already) and then use AJAX for the subsequent links.
For what it's worth, this doesn't sound like a terrible idea to me - especially for bandwidth-sensitive uses like mobile pages. You'd have to carefully weigh the drawbacks to make sure it was worth it in your case, though.