I am wondering if it is still worth with modern compilers and their optimizations to write some critical code in C instead of C++ to make it faster.
no. keep it readable. if your team prefers c++ or c, prefer that - especially if it is already functioning in production code (don't rewrite it without very good reasons).
I know C++ might lead to bad performance in case classes are copied while they could be passed by reference
then forbid copying and assigning
or when classes are created automatically by the compiler, typically with overloaded operators and many other similar cases
could you elaborate? if you are referring to templates, they don't have additional cost in runtime (although they can lead to additional exported symbols, resulting in a larger binary). in fact, using a template method can improve performance if (for example) a conversion would otherwise be necessary.
but for a good C++ developer who knows how to avoid all of this, is it still worth writing code in C to improve performance?
in my experience, an expert c++ developer can create a faster, more maintainable program.
you have to be selective about the language features that you use (and do not use). if you break c++ features down to the set available in c (e.g., remove exceptions, virtual function calls, rtti) then you're off to a good start. if you learn to use templates, metaprogramming, optimization techniques, avoid type aliasing (which becomes increasingly difficult or verbose in c), etc. then you should be on par or faster than c - with a program which is more easily maintained (since you are familiar with c++).
if you're comfortable using the features of c++, use c++. it has plenty of features (many of which have been added with speed/cost in mind), and can be written to be as fast as c (or faster).
with templates and metaprogramming, you could turn many runtime variables into compile-time constants for exceptional gains. sometimes that goes well into micro-optimization territory.