What are the advantages and disadvantages of DTOs from a website performance perspective? (I'm talking in the case where the database is accessed on a different app server to the web server - and the web server could access the database directly.)
DTO's aren't a performance concern. I think what you are asking about is the performance implications of tiering. In particular, using an application tier between your web tier (web server) and data tier (database server).
Generally, the implications are that latency is increased (you have extra network roundtrips), but you gain some additional capacity by splitting the load across machines.
Another common reason (again, non-performance) that people would do that is to allow them to place the web server in the DMZ while keeping the application and database servers inside the firewall.
Another potential reason (non-performance) is the ability to plug multiple UIs on top of a single application. I've done this on past projects with great results (where the business required it).
Also, don't underestimate the work required to maintain an architecture of that nature. It's more work than a non-tiered solution, so only use it if you anticipate needing it.
That being said, the use of DTOs does not necessitate the use of Tiering.
The best description I've found of tiering comes from Martin Fowler's book, Analysis Patterns. There's a small section in the back on application facades and tiering.
Just to reiterate the previous answer, DTOs aren't a performance concern. It's just a class without methods used to provide isolation between various parts of your application.
I'd also suggest picking up Martin's other book, Patterns of Enterprise Application Architecture. The DTO "pattern" is documented there.