There are many LAMP stacks of a few to a dozen of servers around that have serious troubles with performance, troubleshooting and manageability.
To give a clear example. Let's think of a setup with 10 apache servers serving static files and PHP, a mysql database server and a haproxy load balancer in front. Assume that there are 5 sites with different domains load balanced across the servers. Some common problems in a setup like this, and possible solutions are:
Apache performance is mediocre at best.
Most probably 2-4 of the apache servers can be removed/reused for something else by using a well configured lightweight web server. The two most common choices are nginx and lighttpd, in that order. In my opinion they are both stable and pretty equivalent in performance - but I do like the flexibility in nginx configuration a bit better.
No good cache system utilized makes each and every PHP request come with a big overhead.
Changing the plain load balancer into a caching one can come with great performance gains. Varnish is now probably the first choice for most people and is both very stable and performs well. By setting an expiration time from your backend (web) servers you can control exactly how long each file should be cached. Put some time and effort to achieve high hitrates and it is probably the most important change you can do.
No good way to identify bottlenecks or performance troubles in the PHP code. With many recent releases over several sites in the web cluster it's very hard to know the reason for overloaded servers.
Having a live profiler for PHP in your web servers is a great win in this situation. If you want something free there is XHPROF that was open sourced by Facebook. Unfortunately they did not release their web interfaces that made it useful in production. There are however two open source solutions for that Xhgui that logs into a mongodb database and XhProf Analytics that logs into mysql. Just take caution on where and how you use it as it's quick to fill up your disk with the profiler results A good setup could be to log only a percentage of the requests and then to empty it from old data (with an index for mongodb and a cronjob for mysql).
If you don't mind paying a bit there is also a hosted solution called New Relic with a nice intuitive interface. They do more than just PHP as well.
No good way to get an overview of where errors happen. To read the PHP error logs you need to log into each of the machines.
Use a centralized logging service.
If you want to roll your own there's a very good solution in Graylog2 together with Logstash for collection. Graylog2 provides a user interface with search, log streams and alerts and Logstash provides an easy to use client for parsing and forwarding log file messages to the Graylog2 server.
For hosted solutions the most mentioned service is loggly. Beware that it might become costly depending on your logging scale.
MySQL uses MyISAM and it keep causing locks.
With row level locks instead of table level InnoDB mostly comes as a must when you get alot of writes to some hit tables. It will also in general be faster for an average setup with it's buffer pool caching of data, not to mention it's primary key partitioning. Be sure that you have a large enough buffer pool (and suffificent memory for it) and it will do miracles for your database reliability.
Other things to consider
Are you using the right database system?
There are many cases where alot of your database load can be transferred into a more lightweight system. A common situation is that many simple data structures can be moved from a somewhat heavy RDBMS (say MySQL) to a more lightweight NoSQL database such as Redis. This can increase the performance significantly if used in hotspots with simple data structures.
Revisit your database load
Database loads are a very common bottleneck and is most often the hardest part of your application to scale. By periodically checking up on what is causing the load there are usually many easy optimizations to be done. pt-query-digest is an excellent tool for profiling your database load and finding the heavy queries.
Measure your performance. See what causes the most load on the servers with xhprof/new relic, varnish cli tools, pt-query-digest etc. You will surely find things within the top requests and queries that can be more effeciently cached.
This list is far from complete but I think and hope I managed to get the most common and important points down.