How relevant are OO design patterns in developing PHP applications for the web?
IMO they are very relevant for any non-trivial PHP application, as long as the pattern you are using is applicable for the problem at hand.
Does it do anything for performance?
Not necessarily, and you can take that in both the positive and negative respects. Most popular design patterns aren't going to directly add any positive performance benefit, and some that impose more inheritance or class structure might add a negligible performance cost (I emphasize negligible). The benefits of using well-known patterns, when applicable, outweigh these negligible costs.
One example I can think of where a pattern would directly improve "performance" (with regard to memory usage) would be the appropriate use of the singleton pattern. If you really only need one instance of an object at any given time, then you minimize memory usage by using that instance.
Or is it just to keep code lean for agile development practices?
I would say that correct use of patterns keeps code more maintainable, rather than "lean." This would facilitate any development cycle, including agile, since code that uses well-known patterns is easier to read.
Who is the major benefactor for implementing these design patterns? Is it the customer or the developer?
Primarily the developer (especially if it is a plural developer(s) where a team has deal to look at the same code.) There is an indirect impact on the customer in that well-written code will obviously have less bugs, and software that uses patterns will probably have less turnaround time from re-inventing the wheel.