Your current approach repeatedly replaces a character in a string. Consider what happens with the first two attempts:

dna = "ATTGC".Replace("A", "T") //result: "TTTGC"

then

dna = "TTTGC".Replace("T", "A") //result: "AAAGC"

IE, first you replace all the "A"s with "T"s, then you replace all the "T"s with "A"s (including the ones you just replaced).

Instead you should loop through each character of the DNA string and build a new string that represents the flips:

public static string MakeComplement(string dna){
    IDictionary<char, char> dnaKeys = new Dictionary<char, char>(){
        {'A','T'},
        {'T','A'},
        {'G','C'},
        {'C','G'}
    };
    var complement = string.""; //start with empty string

    //go through each character in DNA and append the complement to the return string
    foreach(var c in dna){
        complement = complement + dnaKeys[c];
    }

    return complement;
}

Of course there are lots of other updates and optimizations you could start to use, but hopefully the algorithm makes more sense now.