What you are referring to is a tax lien, where (as you note) investors will pay the municipality the balance owed on the taxes in exchange for a lien against the property. The investor will then attempt to collect the amount owed from the current owner (your tenant) and collect the spread (inclusive of the original amount plus any fees + interest). This is the high level overview.
While I am not familiar with Illinois specifically, the process generally plays out the same anywhere I am aware of. And that is the investor can take over title of the home if the owner (your tenant) does not pay the amount due, becoming the de facto owner. At that point, the previous owner would then become a tenant of the investor and the investor would essentially become a Lonnie Dealer.
The implication, bottom line, is that the investor would owe you the lot rent and be responsible for adhering to the rules of your park (and ensuring his new tenant does the same).
I have never done a tax lien on a mobile home inside of a park and am struggling to visualize the best play for you here. However, you might consider reaching out to the investor and the current owner directly and trying to strike some kind of deal that ensures you keep the home in your park. However, if the investor is doing this specifically to become a Lonnie, that might not be a bad thing for you…