Like user unknown already said, "Irrtümer" are about believing in something which is untrue, "Fehler" means doing things wrong. In everyday life, an Irrtum often leads to a Fehler: If you think your coworker does not need a piece of paper anymore (which is wrong, this would be an Irrtum) and you throw it away, this action would be the wrong thing to do (a Fehler).
Typos are called "Tippfehler"; we call spelling mistakes "Rechtschreibfehler". If a teacher makes corrections to an exam, he/she will mark and count also the number of "Fehler". It does not matter if the "Fehler" are based on an "Irrtum" (e.g. you thinking the river "Rhein" is spelled without the "h"), it is the action of writing the wrong letters which make this a Fehler rather than an "Irrtum".
What about computers, machines and similar items? These can only make "Fehler" (maybe some will argue that artificial intelligences could also have misconceptions, and therefore the term "Irrtum" would be appropriate).
Small defects in materials are called "Materialfehler", deviations in a measurement are "Messfehler".
Jede Messung besitzt/hat einen Messfehler.
A bottle of wine has a corky smell? It has a "Weinfehler". Note the difference in the verb in the last examples, maybe this example helps:
Peter machte einen Fehler: Er vergaß das Messgerät richtig einzustellen. Als Ergebnis hatte die Messung einen großen Messfehler.