I'm responsible for some application level code that I inherited that has some x86... Intel assembly code based on 32 bit instructions and addressing. I going to make the assumption this 32 bit assembly code will run when we migrate a 64 bit windows OS. Yes?
Yes. 32-bit code will run on 64-bit machines, but they won't be able to access more than 4 GiB of RAM.
It will probably run, but it won't take advantage of any of the features the 64-bit extensions to x86 asm provide (x86-64, which is what all "64-bit" versions of Windows are) or be able to tightly interop with x64 libraries. The WoW64 layer is very stable and 99.999% of 32-bit apps run fine on current "64-bit" OSes.
If you want to migrate to a true 64-bit operating system (the only ones I know of are Itanium-based, ia64), you're out of luck.
Edit: Also, if you want to enable large-address access (able to use >4gb of RAM), there's a PE header flag you can flip that will turn that on and allow it to address larger amounts of memory.
It will run if you continue to build 32-bit x86 applications, yet. However if you have to port the code to 64-bit x86, chances are that the assembly code won't run (and might not even compile) out of the box.
And depending on your tool chain and if it's inline assembler in C/C++ code or not, you might not even be able to compile it at all.