I want to make sure everything is as lazy as can be without introducing weird semantic issues stemming from ignorance.
…alright, but remember  you did want “to make sure”.
Now to answer the question:
Is unsafeInterleaveIO
idempotent?
or:
unsafeInterleaveIO (unsafeInterleaveIO m)
≟ unsafeInterleaveIO m
(note the small question mark over the equals symbol).
Now for some equational reasoning:
(original expression on left)
unsafeInterleaveIO (unsafeInterleaveIO m)
(from the definition of unsafeInterleaveIO
)
= unsafeDupableInterleaveIO (noDuplicate >> (unsafeDupableInterleaveIO (noDuplicate >> m)))
…question: what is noDuplicate :: IO ()
?
Ensures that the suspensions under evaluation by the current thread
are unique; that is, the current thread is not evaluating anything
that is also under evaluation by another thread that has also executed
noDuplicate.
This operation is used in the definition of unsafePerformIO to
prevent the IO action from being executed multiple times, which is
usually undesirable.
…so in this situation calling noDuplicate
more than once is redundant, as m
is part of the larger I/O action noDuplicate >> (unsafeDupableInterleaveIO (noDuplicate >> m))
Back to the equational reasoning:
(inner noDuplicate
call redundant)
= unsafeDupableInterleaveIO (noDuplicate >> (unsafeDupableInterleaveIO m))
(assumption: only one I/O thread in use)
= unsafeDupableInterleaveIO (unsafeDupableInterleaveIO m)
Two points:

use assumptions wisely;

the next step would be to use the definition of unsafeDupableInterleaveIO
:
unsafeDupableInterleaveIO (IO m)
= IO ( \ s > let
r = case m s of (# _, res #) > res
in
(# s, r #))
to replace its calls. For larger definitions, sometimes using an auxiliary definition can be helpful:
unsafeDupableInterleaveIO (IO m)
= IO (\ s > unsafeDupableInterleaveIO# m s)
unsafeDupableInterleaveIO# m s =
= let
r = case m s of (# _, res #) > res
in
(# s, r #)
Now all the actual work is done by unsafeDupableInterleaveIO#
.
More equational reasoning:
(change of name: m
→ act
)
= unsafeDupableInterleaveIO (unsafeDupableInterleaveIO act)
(from that modified definition of unsafeDupableInterleaveIO
, and act = IO m
)
= case act of IO m > unsafeDupableInterleaveIO (IO (\ s > unsafeDupableInterleaveIO# m s))
(…again, but taking care to avoid name clashes)
= case act of IO m > IO (\ s0 > unsafeDupableInterleaveIO# (\s > unsafeDupableInterleaveIO# m s)) s0)
(since unsafeDupableInterleaveIO#
is a function)
= case act of IO m > IO (\ s0 > unsafeDupableInterleaveIO# (unsafeDupableInterleaveIO# m) s0
(…and again)
= case act of IO m > IO (unsafeDupableInterleaveIO# (unsafeDupableInterleaveIO# m))
From all of that, your original question now reduces to:
unsafeDupableInterleaveIO# (unsafeDupableInterleaveIO# w)
≟ unsafeDupableInterleaveIO# w
 again, note that small question mark,
 and the new name
w
, to avoid confusion with the original m
(and subsequently act
)
Now to start a second equationalreasoning session:
unsafeDupableInterleaveIO# (unsafeDupableInterleaveIO# w)
=

(since unsafeDupableInterleaveIO#
is a function of two parameters)
unsafeDupableInterleaveIO# (\s > unsafeDupableInterleaveIO# w s)

(from the definition of unsafeDupableInterleaveIO#
)
unsafeDupableInterleaveIO#
(\ s > let
r = case w s of (# _, res #) > res
in
(# s, r #))

(again, since unsafeDupableInterleaveIO#
is a function of two parameters)
\ s0 > unsafeDupableInterleaveIO#
(\s > let
r = case w s of (# _, res #) > res
in
(# s, r #))
s0

(using the definition of unsafeDupableInterleaveIO#
again)
\ s0 > let
r = case (\s > let
r = case w s of (# _, res #) > res
in
(# s, r #))
s0
of (# _, res #) > res
in
(# s0, r #)

(renaming the “outer” r
and res
to avoid confusion later)
\ s0 > let
r1 = case (\s > let
r = case w s of (# _, res #) > res
in
(# s, r #))
s0
of (# _, res0 #) > res0
in
(# s0, r1 #)
From here, it’s just a matter of:

applying that “inner lambda” (\ s > let ... in (# s, r #))
to its argument s0
,

simplifying those case
expressions and changing names as needed,

then carefully observing the final expression, looking for (sub)expressions which can be replaced by a combination of calls to existing functions.
If you’re wondering why I’m leaving these last few steps to you:
…so please don’t “copy & paste” what you did to answer your original question:
Is unsafeInterleaveIO
idempotent?
A simple “yes” or “no” will suffice :D