Does adding optional parameters change method signatures and would it trigger method missing exception?

You don't need to remove and re-add the reference, but you do need to rebuild all of the projects that compile against the DLL. Adding an optional parameter is a source-compatible but not a binary-compatible change, because the compiler emits different IL at the call site, to include the default value as normal - the call itself looks the same in the IL whether the optional parameter was omitted or not.

For example, consider the following code:

class Test
{
    static void Main()
    {
        Foo(3);
        Foo();
    }

    static void Foo(int x = 5)
    {
    }
}

The IL for Main looks like this:

  .method private hidebysig static void  Main() cil managed
  {
    .entrypoint
    // Code size       16 (0x10)
    .maxstack  8
    IL_0000:  nop
    IL_0001:  ldc.i4.3
    IL_0002:  call       void Test::Foo(int32)
    IL_0007:  nop
    IL_0008:  ldc.i4.5
    IL_0009:  call       void Test::Foo(int32)
    IL_000e:  nop
    IL_000f:  ret
  } // end of method Test::Main

As you can see, the constant 5 is loaded in the Main method and passed as an argument, just as 3 is explicitly. The same thing happens across assembly boundaries.

The same rules also applies to changing the values of constants, and changing the default value of an optional parameter - all of these require client rebuilds.