You should generate a unique ID for each element you are adding to the page. You can use a variable and increment this variable each time an element is added. In your exemple, the variable 'patient' is enough.
add.innerHTML += "<div id=removepatient_"+patient"></br>Patient "+patient+" Name/ID:<input type=text name=patients[] ><input type=button id=more_fields"+patient+" onclick=removepatient("+patient+"); value='Remove' ></div>";
You must also add a param to your delete function :
function removepatient(patient) {
var elem = document.getElementById('removepatient_'+patient);
elem.parentNode.removeChild(elem);
return false;
}
I hope it helps you