Precautions in designtime to consume less memory... (you would watch the vb6.exe memory consumption in debug mode...)
Use control arrays as much as possible.
Don't over crowd forms with controls.
If you can, put as much code into modules and classes as you can as the forms graphical elements take up a lot of memory.
When using data access methods and you have a table with an extreme amount of records, don't use a select * unless you absolutly have to for your program.
When you can unload formname/set formname = nothing to totaly remove it from memory.
And this goes with the one above... Don't use public variables in the forms because if you need access to the information contained within that variable, you have to keep at least part of the form loaded. So put the public variable in a module and that way you will have access to it from everywhere when you need it.
Don't load an excessive amount of graphics into memory (icons, who cares, but pictures like bmp's, jpeg's, and the like, yeah you gotta watch those).
The same thing can be said about files you open up, meaning text files and such that you open with the Open Method.
When you can, use an image control instead of a picturebox control or a label instead of a textbox... i.e. use the lightest weight control you can for the purposes needed at hand...