The dead code problem is related to the Halting problem.
Alan Turing proved that it is impossible to write a general algorithm that will be given a program and be able to decide whether that program halts for all inputs. You may be able to write such an algorithm for specific types of programs, but not for all programs.
How does this relate to dead code?
The Halting problem is reducible to the problem of finding dead code. That is, if you find an algorithm that can detect dead code in any program, then you can use that algorithm to test whether any program will halt. Since that has been proven to be impossible, it follows that writing an algorithm for dead code is impossible as well.
How do you transfer an algorithm for dead code into an algorithm for the Halting problem?
Simple: you add a line of code after the end of the program you want to check for halt. If your dead-code detector detects that this line is dead, then you know that the program does not halt. If it doesn’t, then you know that your program halts (gets to the last line, and then to your added line of code).
Compilers usually check for things that can be proven at compile-time to be dead. For example, blocks that are dependent on conditions that can be determined to be false at compile time. Or any statement after a return (within the same scope).
These are specific cases, and therefore it’s possible to write an algorithm for them. It may be possible to write algorithms for more complicated cases (like an algorithm that checks whether a condition is syntactically a contradiction and therefore will always return false), but still, that wouldn’t cover all possible cases.