Python 2, 43 bytes

lambda s,*l:eval('s'+'.replace(*l)'*len(s))

Try it online!

Evaluates a string of the form

s.replace(*l).replace(*l).replace(*l) ...

To reach a fixed point if one exists, it suffices to do replacements equal to the length of the original string.

ES6 (Javascript), 47, 43 bytes

• Saved 4 bytes using currying (Thanks @Neil !)

Golfed

c=>b=>R=a=>(x=a.split(b).join(c))==a?x:R(x)

Try It

Q=c=>b=>R=a=>(x=a.split(b).join(c))==a?x:R(x)

function doit() {
  console.log(Q(C.value)(B.value)(A.value));
}

A: <input type="text" value="abaaba" id="A"/> B: <input type="text" value="aba" id="B"/> C: <input type="text" value="ab" id="C"/> <input type="submit" onclick="doit();" value="REPLACE"/>

Retina, 27 bytes

Byte count assumes ISO 8859-1 encoding.

+`(.+)(?=.*¶\1¶(.*))
$2
G1`

Input should be linefeed-separated.

Try it online! (For convenience, uses a test-suite input format where each line is a slash-separated test cases.)

C#, 44 Bytes

Short Version:

r=(a,b,c)=>a==(a=a.Replace(b,c))?a:r(a,b,c);

Example Program:

using System;

namespace ConsoleApplication1
{
    class Program
    {
    static void Main(string[] args)
        {
            Func<string, string, string, string> r = null;
            r=(a,b,c)=>a==(a=a.Replace(b,c))?a:r(a,b,c);

            Action <string, string, string, string> test =
                (a, b, c, answer) =>
                {
                    var result = r(a, b, c);
                    Console.WriteLine("A: \"{0}\"\r\nB: \"{1}\"\r\nC: \"{2}\"\r\nResult: \"{3}\"\r\n{4}\r\n\r\n",
                        a, b, c, result, result == answer ? "CORRECT" : "INCORRECT"
                        );
                };

            test("Hello, world!", "world!", "PPCG", "Hello, PPCG");
            test("Uppercase is up", "up", "down", "Uppercase is down");
            test("ababababa", "aba", "ccc", "cccbcccba");
            test("delete", "e", "", "dlt");
            test("", "no", "effect", "");
            test("llllrrrr", "lr", "rl", "rrrrllll");
            test("+-+-+-+", "+-+", "+", "+");
            test("ababababa", "aba", "bada", "badabbadbada");
            test("abaaba", "aba", "ab", "abb");
            test("((())())())", "()", "", ")");


            Console.WriteLine("Press any key...");
            Console.ReadKey();
        }
    }
}

Explanation: The function is written as a tail recursive expression, avoiding the return keyword and curly brackets by exploiting the following:

• An assignment within parenthesis returns the value assigned
• The left side of the equality check will be evaluated before the right side assignment, allowing us to compare before/after inline, and still access the result

This lets us keep it to a single statement.

EDIT: Went back to omitting the type of function r, since that appears to be acceptable. With type declaration using arrays, it is 68 characters. Without, it is 44 characters.

Ruby, 29 bytes

->a,b,c{1while a.gsub! b,c;a}

Given 3 arguments, apply substitution to the first until there is nothing to substitute anymore.

Explanation

• 1 before the while is simply a nop
• gsub! returns the string or nilif no substitution occurred

Japt, 15 bytes

@¥(U=UqV qW}a@U

Test it online!

How it works

@¥(U=UqV qW}a@U  // Implicit: U, V, W = input strings
            a@U  // Return the first non-negative integer mapped by the function X => U
@          }     // that returns truthily when mapped through this function:
     UqV qW      //   Split U at instances of V, and rejoin with W.
  (U=            //   Set U to this new value.
 ¥               //   Return (old U == new U). This stops the loop when U stops changing.
                 // Implicit: output result of last expression

Japt has a recursive-replace built-in, but it sees the first input as a regex. If the inputs were guaranteed to only contain alphanumeric characters this three-byte solution would work:

eVW

If the input were allowed to contain any char except ^, \, or ], this 12-byte solution would be valid instead:

eV®"[{Z}]"ÃW

C#, 33 49 bytes

Probably, one of the smallest snippets written in C#... And since Replace is native to the string struct, there's no need for usings ( At least not on VS built-in feature, C# Interactive... )

Also, since B always has a value, the code doesn't need any validations.

Golfed

(a,b,c)=>{while(a!=(a=a.Replace(b,c)));return a;}

Ungolfed

(a, b, c) => {
    while( a != ( a = a.Replace( b, c ) ) );

    return a;
}

Full code

using System;

namespace Namespace {
    class Program {
        static void Main( string[] args ) {
            Func<string, string, string, string> func = (a, b, c) => {
                // Recursively ( now truly recursive ) replaces every 'b' for 'c' on 'a',
                // while saving the value to 'a' and checking against it. Crazy, isn't it?
                while( a != ( a = a.Replace( b, c ) ) );

                return a;
            };

            int index = 1;

            // Cycle through the args, skipping the first ( it's the path to the .exe )

            while( index + 3 < args.Length ) {
                Console.WriteLine( func(
                    args[index++],
                    args[index++],
                    args[index++]) );
            }

            Console.ReadLine();
        }
    }
}

Releases

• v1.1 - +19 bytes - Fixed solution not being recursive.
• v1.0 -  33 bytes - Initial solution.

Processing, 75 72 bytes

void g(String a,String[]s){for(;a!=(a=a.replace(s[0],s[1])););print(a);}

Prints the results. Call it like g("llllrrrr", new String[]{"lr","rl"});

void Q110278(String a, String[]s){             //a is the string to be replaced
                                               //s is the array containing the subsitution

  for(; a!=                                    
            (a = a.replace(s[0], s[1])) ;);

  //for-loop where we continuously perform substitution on a
  //until a is equal to substituted a


  //at the end, print the final version of a
  print(a);
}

Mathematica, 35 32 Bytes

#//.x_:>StringReplace[x,#2->#3]&

Arguments given as a sequence. Never terminates for grow example, returns loop for loop example. Three bytes off thanks to Martin's suggestion.

Pyke, 6 bytes

hVQtX:

Try it here!

///, 3 bytes

///

Put string B after the first slash, C after the second and A at the end, ie:

/<B>/<C>/<A>

Try it online!

JavaScript (Firefox 48 or earlier), 43 bytes

c=>b=>g=a=>a==(a=a.replace(b,c,'g'))?a:g(a)

Takes arguments curried in reverse order. Firefox used to have a non-standard third parameter to replace which specified regexp flags. This parameter was removed in Firefox 49.

SmileBASIC, 72 68 bytes

I=0DEF R A,B,C
I=INSTR(A,B)?A*(I<0)A=SUBST$(A,I,LEN(B),C)R A,B,C
END

One of the rare cases where making a function is actually SHORTER in SmileBASIC.

Javascript 130 bytes

f=(a,b,c)=>a.indexOf(b)<0?a:f(eval("a.replace(/"+b.replace(/([\/\,\!\\\^\$\{\}\[\]\(\)\.\*\+\?\|\<\>\-\&])/g,"\\$&")+"/g,c)"),b,c)

Javascript will only replace all simultaneously if you give it a regex. In order to make this regex work for all values, all characters that are used for regex need to be replaced with the escaped version. Finally, the replace is evaluated to replace all instances of B in A with C and passing that back around to the function again.

q, 15 bytes

{ssr[;y;z]/[x]}

Example:

q){ssr[;y;z]/[x]}["llllrrrr";"lr";"rl"]
"rrrrllll"

link to interpreter download

Explanation: ssr, / (converge)

Cheddar, 37 bytes

(a,b,c)f->a has b?f(a.sub(b,c),b,c):a

On phone so TIO link is a bit difficult to add. Basically uses recursion while checking is b is in a. Solution could of been (a,b,c)->a.sub(Regex{b,"cr"},c) but doesn't work for some reason.

Perl 6, 40 bytes

{$^b;$^c;($^a,{S:g/$b/$c/}...*eq*)[*-1]}

Try it (if tio.run gets updated)
Try an altered version

Expanded:

{
  $^b;           # declare second parameter ( not used here )
  $^c;           # declare third parameter  ( not used here )

  (

    $^a,         # declare first parameter, and use it to seed the sequence

    {S:g/$b/$c/} # replace globally

    ...          # keep doing that

    * eq *       # until there are two that match

  )[*-1]
}

PHP, 46 bytes

function g($a,$b,$c){echo strtr($a,[$b=>$c]);}

PHP, 102 bytes

list($n,$s,$a,$b)=$argv;$f=str_replace($a,$b,$s);while($s!=$f){$s=$f;$f=str_replace($a,$b,$s);}echo$f;

Test cases (functional)

Test case with loop error

Java - 157 bytes

String r(String f){if(f.length()<1)return "";String[]x=f.split("/");if(x[0].contains(x[1]))return r(x[0].replace(x[1],x[2])+'/'+x[1]+'/'+x[2]);return x[0];}

For empty input it returns an empty string.

Crashes with StackOverflowException error when B is empty or it is fed with data like this A/A/A.

How it works:

r("ABCD/A/F") returns value of r("FBCD/A/F") which returns FBCD
If there is no more characters to be replaced it returns the final output

Ungolfed code code with comments:

String r (String f) {
    if(f.length() < 1)
        return ""; // For empty input return empty output
    String[] x = f.split("/"); // Get all 3 parameters
    if (x[0].contains(x[1])) // If input contains replaced value
        return r(x[0].replace(x[1],x[2])+'/'+x[1]+'/'+x[2]); // Return value of r() with one character replaced
    return x[0]; // If nothing to replace return the output(modified input)
}

AutoHotkey, 87 bytes

StringCaseSense,On
Loop{
IfNotInString,1,%2%,Break
StringReplace,1,1,%2%,%3%
}
Send,%1%

%1%,%2%, and %3% are the first 3 arguments passed to a function
If a function expects a variable argument, the %s are dropped
Changing the case-sensitivity setting costs 19 bytes but, without it, you get things like downpercase is down.