If your application has been developed and is running slow, then design patterns wouldn't be my first point of call. I would look at the SQL queries as this has been the bottle neck for me a lot of times.
Also try to make the application grab chunks of data and implement caching as one of the other replies has already stated.
As for design patterns, this is a very fluffy question that doesn't really have a correct answer. If I were you I would look at the Enterprise Library and either the Smart Client Software Factory or the web equivanent depending on what platform you are targeting.
Some patterns you may want to reseach are:
Repository Pattern
Data Transfer Objects
Data Mapper Pattern
You may also want to look at other example 3 tier applications and get a feel of what patterns they are using and whether these would work for you. I would recommend the Timok exaple (google it).
I hope this helps...
B