Should I use const for local variables for better code optimization?

This performance difference will almost certainly be negligible, however you should be using const whenever possible for code documentation reasons. Often times, compilers can figure this out for your anyway and make the optimizations automatically. const is really more about code readability and clarity than performance.

Or am I wrong, and compilers can figure out by themselves that the local variable is never modified?

Most of the compilers are smart enough to figure this out themselves.
You should rather use const for ensuring const-correctness and not for micro-optimization.
const correctness lets compiler help you guard against making honest mistakes, so you should use const wherever possible but for maintainability reasons & preventing yourself from doing stupid mistakes.

It is good to understand the performance implications of code we write but excessive micro-optimization should be avoided. With regards to performance one should follow the,

80-20 Rule:

Identify the 20% of your code which uses 80% of your resources, through profiling on representative data sets and only then attempt to optimize those bottlenecks.