I know that "its" is the possessive form of "it", but does this rule apply to the possessive form of phrases ending in "it"? Should I say, "the program runs on whomever runs its computer" or "the program runs on whomever runs it's computer" (also I hope I'm right that that should be "whomever" and not "whoever")?
NOTE: Ignore this first bit and skip down to the EDIT section for the right answer. I misread the sentence on first (and second, and third) reading.
The closest sentence with correct grammar (but not sense) that matches your own is:
The program runs on whoever runs its computer.
Because the object of the preposition on is not **whomever*, which is wrong, but rather the entire clause whoever runs its computer, in which whoever acts as the subject of the verb runs. The proximity of on is a red herring — or strange attractor, depending on which metaphor you prefer.
Since you can replace its with his — and not with it is — this shows that its must be correct.
However, all that said and done, I am unconvinced that this sentence makes sense. I bet you want:
The program runs on the computer of whoever runs it.
Because I don’t think it runs on people, but rather on computers.
Maybe you could almost get away with:
The program runs on whosever computer runs it.
But that seems a bit off, too.
Now I see what you were going for! You were trying to use the Saxon genitive on the entire noun phrase:
The program runs on (whoever runs it)’s computer.
Which is just like:
The program runs on (the person who runs it)’s computer.
That way it would work the same way as these do:
Unfortunately, without parentheses, people will too easily misread it, as in fact did I the first few times through it.
While there is certainly some sense in what you say, it if nothing else garden-paths to the wrong it’s, which makes it look like a mistake even when it isn’t.
I’d still say it runs on the computer of whoever runs it myself, just to make sure.
Perhaps this will help with deciding what case to use for the relative pronoun. Whether or not a relative pronoun should be in the objective or in the nominative case is determined by the relative pronoun's function within its relative clause. Some examples:
the girl who loves me
Here, who is the subject of the verb loves in the relative clause who loves me; therefore one uses the nominative case, who, not the objective case, whom. It doesn't matter what is the function in the sentence of the relative pronoun's referent, girl . Thus,
The girl who loves me loves another as well.
I love the girl who loves me.
In the first example the referent is the subject of the verb of the sentence; in the second it is the object of the verb love. It doesn't make any difference.
the girl whom I love
Here, whom is the object of the verb love in the relative clause whom I love; therefore one uses the objective case, whom, not the nominative case, who. It doesn't matter what is the function in the sentence of the relative pronoun's referent, girl . Thus,
The girl whom I love loves another.
Another loves the girl whom I love.
In the first example the referent is the subject of the verb of the sentence; in the second it is the object of the verb loves. Again, it doesn't make any difference.
Some people get a little baffled when, instead of noun referents, there are pronoun referents, but it still works the same way:
The girl who loves me loves another as well. >> She who loves me loves another as well.
I love the girl who loves me. >> I love her who loves me.
The girl whom I love loves another. >> She whom I love loves another.
Another loves the girl whom I love. >> Another loves her whom I love.
But what really confuses people is when the relative pronoun has no explicit referent, which has been omitted by a kind of ellipsis, as it were:
Whoever loves me loves another as well.
I love whoever loves me.
Whomever I love loves another.
Another loves whomever I love.
In such cases, we say that the entire relative clause functions in the sentence as its relative pronoun's absent pronoun referent would have. This confuses people who then try to put the relative pronoun in the same case that its "missing" pronoun referent would have had.