64-bit alone will do a little bit for performance. Raising the game from SSE to SSE2 will allow the compiler to do better optimizations, especially for mathematical calculations (as well as more consistent rounding). And 64-bit compilers have new rules that finally force them to remove a bunch of legacy code that is incompatible with some compiler optimizations. It also forces more proper memory alignment in the engine, which can improve performance.
On Windows, 64-bit will eliminate the WoW64 compatibility layer, which adds some overhead to threading and disk reading mainly. I am not sure of the overhead, if any, of 32-bit support on Linux.
64-bit also increases the amount of RAM available from 4GB to practically unlimited. This fixes out-of-memory crashes that people sometimes get, especially with higher texture qualities (on DX8, I believe, and on some Linux drivers) because of system copies of texture data. But it also happens for 100-player servers and custom MvM missions sometimes.
The increased RAM can also allow people to increase cache sizes, which can help with cosmetic and sound loading and the stutters associated with both.
Furthermore, on Linux, the runtime is being upgraded from Ubuntu 12.04 to Steam Linux Runtime 3 (sniper), a much more modern set of tools in terms of the compiler, SDL graphics/input library, and more. This has performance improvements, bug fixes (for example, the MvM TFBot name issue on Valve servers), console text rendering fixes, and a double mouse sensitivity fix.
The port also means some libraries are being changed. The memory allocator is being changed to mimalloc, which is much faster and means there will be less overhead for a lot of operations. The MP3 reading is being changed from Miles to minimp3, which is also faster.
A new dxvk-based layer for DirectX to Vulkan is being added, which gets graphical parity for Linux compared to Windows, and also gets rid of overhead, massively improving performance for graphics rendering on Linux. On Windows, the dxvk results have been quite middling and inconsistent, though.
Finally, performance changes are being made, from TC2 and some new fixes. The extent of this is not fully known or realized yet but I can say that I am very optimistic about some pretty impactful performance improvements getting in.