Should a web server care about IPv6 compatibility?

If you're running a small in house server for you company for their computers, you won't probably need it.

For the rest of the web, IPv6 is something happening right now and you NEED to support IPv6, because there are many and many people joining to the IPv6 network and the transition between the two represent lose of service and delays, so yes you server really need to get an IPv4 and a IPv6 address. IPv6 is also great in other conditions like in media stream because you can just send one packet to multiple clients at the same time instead of sending the same content repeatedly for multiple clients — Less bandwidth needed to live stream.

For instance in my country, Portugal, out brand new optic fiber network is covering 90% of the country and it's fully IPv6... So I guess that other countries are doing the same now. There are no IPv4 addresses—we ran out of them ;)

Of course there are. I actually have IPv6 access at home :) Surely the traffic levels are low currently, but if you're a server admin, you should really consider enabling IPv6 (and all relevant stuff like AAAA records on your site hostnames). IANA has allocated all free IPv4 space to the regional internet registries and thus the allocation of IPv4 space will come to an end soon. A lot of major internet companies are currently testing with IPv6, or already have it enabled on some services. On Wednesday june 8th 2011 some major companies like Google and Facebook will enable IPv6 on their hosts. It'll be a testcase to see if and how stuff breaks. See http://www.worldipv6day.org/ for more info ;)