What are the restrictions on distributing configure
scripts that Autoconf generates? How does that affect my
programs that use them?
There are no restrictions on how the configuration scripts that Autoconf produces may be distributed or used. In Autoconf version 1, they were covered by the GNU General Public License. We still encourage software authors to distribute their work under terms like those of the GPL, but doing so is not required to use Autoconf.
Of the other files that might be used with configure, config.h.in is under whatever copyright you use for your configure.ac. config.sub and config.guess have an exception to the GPL when they are used with an Autoconf-generated configure script, which permits you to distribute them under the same terms as the rest of your package. install-sh is from the X Consortium and is not copyrighted.