It's safe from exhaustion, most likely
First of all, going without sleep causing exhaustion is already a variant rule of sorts, introduced in Xanathar's Guide to Everything (p. 78), which specifically states that you "can use these rules if you want to account for sleep deprivation". I assume they removed the "you don't suffer exhaustion" line to keep it from being confusing, because anybody who doesn't have Xanathar's would be hard-pressed to locate the rule where you get exhaustion.
The optional rule on going without a long rest states:
A long rest is never mandatory, but going without sleep does have its consequences. If you want to account for the effects of sleep deprivation on characters and creatures, use these rules.
Whenever you end a 24-hour period without finishing a long rest, you must succeed on a DC 10 Constitution saving throw or suffer one level of exhaustion.
It becomes harder to fight off exhaustion if you stay awake for multiple days. After the first 24 hours, the DC increases by 5 for each consecutive 24-hour period without a long rest. The DC resets to 10 when you finish a long rest.
While the text seems to make a bit of a mess of the general theme of it, considering it says "whenever you end a 24 hour period without finishing a long rest", the general theme here is sleep deprivation. Your warforged does not require sleep, so they will not have to fight sleep deprivation, there's no real sensible reason to apply a rule "to account for sleep deprivation" on a character or creature that does not need sleep.
If you use the rule, pure RAW, your Warforged will get exhausted because they are sleepy despite not needing to sleep. Reading the entire text however should, in my opinion, make it fairly clear that a creature that does not require sleep will not have to fight against sleep deprivation.
I mean, talking pure RAW, you could sleep 7 hours, walk 1 hour, sleep 7 hours, walk 1 hour, sleep 7 hours, walk one hour and you're now suffering from sleep deprivation because you didn't finish a long rest.
In the end, this is an optional rule so it's entirely up to your DM how it works, but logic suggests your Warforged should be a-ok.