Are there any security issues associated with port-based VLANs?  In
particular, if I want to connect some machines in a DMZ and some
others behind a firewall, with a router in between, to one physical
switch, will using two port-based VLANs for the two groups of machines
compromise the security-related efforts taken to isolate the two sets
of machines?

You have to check the vulnerabilities in the switch, to check if there
are related vulnerabilities to the VLAN (or you particular setup).
There are vulnerabilites associated with trunking and VLAN, under
certain conditions one machine could send TAG (802.11Q acting as
another switch), and jump over other VLAN. If everything is setup
properly yo will have no problem at all. Try to imagine every machine
have the capabilities to talk VLAN-tags like a switch and keep the
setup to protect from this.