What performance tips can be offered to someone running a LAMP server?
In the instance that something is Distribution specific, I'm targeting Debian.
It really depends on your workload.
for the L part
for the A part of your stack - well maybe you want to replace it completely with nginx or lighthttpd, or maybe just leave Apache for dynamic content and have separate server (like those two or mathopd) for static content. Take a look here for more options. If you are going to run both Apache and another server at the same box, a 2nd IP address will be handy. To decrease latency for the end-user use http/1.1 with keep-alive. Consider using a CDN for static content.
for the M part of your lamp - take a look at mysqlperformanceblog. from the top of my head:
for P
Not really a performance tip, but do take offsite backups. Really.
I really suggest separating MySQL and Apache/PHP on two different machines.
For example, I had one machine (C2D E6600) that always spiked to 2.0 and above load average. I put the MySQL on a second machine (P4C 3Ghz) and after that both load averages did not go above 0.2-0.3. So I went from a really slow site to a fast site with two servers having a lot of performance margin.
For the P part, you could consider opcode caching with i.e. APC. One could also consider mod_fastcgi with php instead of the default mod_php.