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If you're concerned just about the static compilation step, and not the JIT, this is simple to check by comparing the bytecode generated from two different classes, using javap:

class WithLocalVar {
 
 private static int methodWithSideEffect() {
 System.out.println();
 return 42;
 }
 public static void main(String[] args) {
 int result = methodWithSideEffect();
 }
}
class WithoutLocalVar {
 
 private static int methodWithSideEffect() {
 System.out.println();
 return 42;
 }
 public static void main(String[] args) {
 methodWithSideEffect();
 }
}
✗ javac With*
✗ javap -c WithLocalVar
Compiled from "WithLocalVar.java"
class WithLocalVar extends java.lang.Object{
WithLocalVar();
 Code:
 0: aload_0
 1: invokespecial #1; //Method java/lang/Object."<init>":()V
 4: return
public static void main(java.lang.String[]);
 Code:
 0: invokestatic #4; //Method methodWithSideEffect:()I
 3: istore_1
 4: return
}
✗ javap -c WithoutLocalVar
Compiled from "WithoutLocalVar.java"
class WithoutLocalVar extends java.lang.Object{
WithoutLocalVar();
 Code:
 0: aload_0
 1: invokespecial #1; //Method java/lang/Object."<init>":()V
 4: return
public static void main(java.lang.String[]);
 Code:
 0: invokestatic #4; //Method methodWithSideEffect:()I
 3: pop
 4: return
}

Therefore, no, the compiler won't optimize away the istore_1. The JIT is another story...