Just signed up to share some of the solutions (3 of them) I was able to come up with in 6 steps:
OH MAN! After a lot of searching I was able to do it in 5 steps (5 solutions):
I confirmed the words on Anagrammer.
EDIT: although all words are accepted on Anagrammer, someone pointed out in the comments that two of the words used in the 6-step solutions ...
Here's one way. It probably isn't close to optimal. 42 steps if I've counted right.
Credit where due: the path from EIGHT to SEVEN is derived from Hunter's answer to an earlier puzzle, though not much of that answer remains in what I have there now. JonMark Perry spotted what in hindsight should have been an obvious improvement in the path from FOUR to ...
This is a proof that Omega Krypton's answer of 38 is optimal, as long as these words are nonexistent (single implication):
First, a calculation of the theoretical best (although no suitable words may exist), by using a Levenshteins distance (word difference by letters) ...
In this answer I will show that five steps is the minimum number required (from the answer by @JoãoBravo).
Suppose by contradiction that there are four. Then the sequence will be of the form
F i v e, _ _ _ _, _ _ _ _, _ _ _ _, F o u r.
If the first letter remains unchanged the whole way through, the sequence is
F i v e, F _ _ _, F _ _ _, F _ _ _, F o u r....
As @RossMillikan pointed out correctly, the proof from @TheSimpliFire is incomplete.
The following part
F i v e, F _ _ _, F _ _ _, F _ _ _, F o u r.
Therefore, at each step, one of i v e must be changed to match o u r, without any deviations. This is clearly impossible.
can be done with any of the following sequences
Five => F_ve => F_ue / F_vr => F_ur ...
Starting with this list of 4 letter words
I've found the following three words that are one away from IRON,
Unfortunately, the first one only leads back to iron or the 2nd word.
The best I've figured using the 2nd word has 7 steps, and uses some uncommon words.
However, using the 3rd word, I've found the following 5 steps, which appears to be optimal.