So the Find Familiar spells says that the caster can see and hear through the familiar's eyes and ears. I'm wondering what happens if the familiar encounters an illusion, say a Minor Illusion, when this is happening. Who makes the intelligence check to disbelieve the illusion, the familiar or the caster? Or do they each get a check?
They both make checks if needed
Find familiar indicates the caster sees through the eyes of the familiar, so the caster is observing and hearing possible illusions:
Find Familiar, PHB pg 240.
Additionally, as an action, you can see through your familiar's eyes and hear what it hears until the start of your next turn, gaining the benefits of any special senses that the familiar has.
The familiar continues to see and operate on its own; you can give it telepathic instructions to fly that-a-way, or "That's not real!".
So the familiar too sees, hears, and smells the illusion, although with such low intelligence, the DM may rule it probably won't use an action to disbelieve it unless the wizard tells it to (exceptions maybe for pseudodragons, quasits, and the like).
Likewise, both caster and familiar would have to make saving throws if in range observing a medusa.
There are no saving throws to disbelieve illusions
How illusions are penetrated is described in each spell description - there is no general rule.
For example, Fear requires a Wisdom saving throw which would be made by the familiar as that would be the creature targeted. In contrast, Major Image and Minor Illusion are detected either by physical interaction (by the familiar) or by using an action to make an Intelligence (Investigation) check (by you or the familiar, whoever is best at it - probably you). And Mirage Arcane requires a creature (the familiar) to have truesight to see the illusion.
In addition, what happens when you have determined that something is an illusion is detailed in the description. For example, Fear just stops, Major Image and Minor Illusion become faint and you can see through them but Mirage Arcane continues to affect you even if you do have truesight.
Even knowing a thing is an illusion doesn't mean that you don't believe in it - think of it as the difference between the intellectual and the emotional: your brain knows it's not really a dragon but your legs are having nothing to do with that! You also can't see through illusions that you have cast yourself unless and until you meet the criteria in the description. Clearly, having cast the illusion or otherwise having strong evidence that it is an illusion should give you advantage on the Intelligence (Investigation) check, if any.
As said in the Player's Handbook Minor Illusion spell description:
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 DC. If the creature discerns the illusion for what it is, the illusion becomes faint to the creature.
Although the tooltip further says that
"Physical interaction with the object reveals it to be an illusion, because things can pass through it."
So I guess that if you can interact with the the illusion (touch it) then you don't need any check, but if you are only seeing or hearing it, then you need the investigation check.
So answering your question, as both the familiar and the familiar's caster are seeing the illusion, I think that both would qualify for the investigation check.
Checking illusions requires an action to use an investigation check, but if I recall, you have to use your action to maintain the sense-sharing, so it'd have to be the familiar checking.