I'd replace

bool result;
if(string.Compare(reversedString, phrase) == 0)
{
    result = true; 
}
else
{
    result = false;
}

return result;

with:

 return string.Compare(reversedString, phrase) == 0;

Other than that, what everyone else has said really: use a regex with ToLower

Also with the code:

if (output == true)
{
    Console.WriteLine("Your phrase is a Palindrome, press enter to continue.");
    Console.ReadLine();
    Console.Clear();                   
}

if (output == false)
{
    Console.WriteLine("Your phrase is not a Palindrome, press enter to continue.");
    Console.ReadLine();
    Console.Clear();                   
}

the only difference is the word 'not', so think about a way to trim that code down.

Also, you do not need to use

if (output == true)

you can you

if (output)

and

if (!output) 

for a negative check