As noted above, live HTML updating is disabled when using your own custom local server.
The Live Preview lightning bolt icon should be also colored red and have a tooltip of "Live Preview (not updating due to syntax error)" in this case, but there's a known bug being tracked as issue #7126 where this sometimes doesn't happen.
Rebooting has resolved many odd issues with Live Development.
On Windows, you may run into issues starting Live Preview if you installed Chrome after installing Brackets.
In order to make it work with Internet Explorer 11, disable all the options that IE uses to include sites in the local intranet (unchecked all the items at to quickly see if the problem is being caused by an extension.
The Theseus and CSS Shapes Editor extensions are known to cause problems with Live Preview, and other extensions could potentially interfere also.