In our present campaign we were fighting mummies. A character (Bard) wanted to cast minor illusion to distract or slow the mummy with an illusion of fire. Would this affect a mummy?
Of course it would!
Of course it is happening inside your head, Harry, but why on earth should that mean that it is not real!
Albus Dumbledore, Harry Potter and the Deathly Hallows by J.K. Rowling
People seem to have a strange little switch that makes them think that if a spell is from the illusion school somehow that means it doesn't do what it says on the box. No one does this with divination or enchantment.
A spell does exactly what it says it does; no more, no less. For a Minor Illusion used this way the relevant parts are:
The image can’t create sound, light, smell, or any other sensory effect. Physical interaction with the image reveals it to be an illusion, because things can pass through it.
If a creature uses its action to examine the sound or image, the creature can determine that it is an illusion with a successful Intelligence (Investigation) check against your spell save DC. If a creature discerns the illusion for what it is, the illusion becomes faint to the creature.
A creature confronted by flames that give off no sound, light or heat is going to do the sensible thing and take a moment to think "Hmmm, that's interesting" and use their free object interaction to physically interact with them to determine that they are not real without breaking stride.
If the character had chosen to create the illusion of something more sensible like, say, a pit, a wall or a staircase, this would require the use of an action to make the investigation check described.
In addition, a creature's type is only relevant if the spell is explicit about it (Monster Manual p. 6):
Certain spells, magic items, class features, and other effects in the game interact in special ways with creatures of a particular type. The game includes the following monster types, which have no rules of their own.
Because Minor Illusion says nothing about creature types they are irrelevant to it. Contrast this with, say, Cure Wounds which is explicit that it doesn't affect Constructs or Undead.
Similarly, there is nothing in the description of the Mummy that indicates any resistance or susceptibility to illusions in general or Minor Illusion in particular.
The Mummy would definitely see the illusion and believe it real, as any other creature would. So in that respects it would "affect" the mummy. "Would it 'distract or slow the mummy' the mummy?" And the answer to that is "Maybe."
Mummies, unlike zombies, aren't just killing machines. They can (but rarely do) speak, and while they are driven by the orders given to them by the one who created them, they can make free choices in the absence of command or in details not in the command.
MM says (p.315):
Zombies take the most direct route to any foe, unable to comprehend obstacles, tactics, or dangerous terrain. A zombie might stumble into a fast-flowing river to reach foes on a far shore, clawing at the surface as it is battered against rocks and destroyed. To reach a foe below it, a zombie might step out of an open window. Zombies stumble through roaring infernos, into pools of acid, and across fields littered with caltrops without hesitation.
Zombies are basically mindless kill machines. Mummies lack this passage, meaning they would take reasonable steps to avoid harm if it doesn't interfere with their given purpose.
If the mummy thought there was a fire in front of them, they would likely move around it -- where a zombie would not.
Driven to Obey at Cost of Own (Un)Life.
That said, the mummy is driven first to "punish transgressors," and would put itself into harms way if it were the only way to do so. That means if you block its path with a fire illusion in the only space between it and you, it will risk the fire damage to come after you.