Let me frame it this way..
"Say I have an application server running on a single core Intel processor & serving 200 concurrent users. Will moving the hardware to a dual core system enable my application server to now serve say 350 concurrent users" ?
The base question I'm hoping to address is - "Can addition of extra processor cores help increase the I/O capabilities of the server". Because sometimes thats exactly what hardware vendors hint at when touting their multi-core servers.
You have to profile your server, determine its bottleneck, and treat them.
If you are memory bound, buy more memory or buy a computer with faster memory.
If you are disk bound, and doing lots of serving static content, consider an SSD drive.
If its your database bound, consider that.
But usually you're "configuration bound", and tweaking your configuration e.g. changing the number of threads in a pool, or such, will give you immediate improvement.
Many webservers and databases are exactly the software written to take advantage of extra cores. Of course, many also charge licensing fees per core, so consider that aspect too.