According to Nehmo Sergheyev :

Anyway, sure, it is ideally preferable to have an individual branch
circuit for every major appliance. But OP cited "$$$", so economy takes
priority. There's a big difference in cost here.

Electrically, it would probably (but not necessarily - see 4) work, but
code takes precidence. Unless you're feeling lucky.

How does this violate code? Let us count the ways:

1) Old-style 3-wire dryer circuits: have a combination ground and neutral.
Running a 120V circuit (or indeed, ground on _anything_ other than a 3-wire stove
or dryer) from a combined ground and neutral is a code no-no and potentially
lethal - that's why 3-wire stove and dryer circuits are "old style" and are
prohibited in new construction.

[In fact, if you rework the circuit, code-wise you need to upgrade to 4wire.
Might as well run a new 120V circuit instead of screwing with the 240V circuit.]

2) New-style 4-wire dryer circuits: simply tapping off a 120V circuit means
that you have what's tantamount to a 15A or 20A 120V circuit breakered at 30A.
Code no-no.

Secondly, this is being used as a "multi-wire" branch circuit. Code prohibits
mixtures of 240V and 120V devices on the same multi-wire branch (the dryer
is treated as a single device obviously).

Thus, regardless of 3 or 4 wire, it still violates code, so asking which
it is is irrelevant.

3) To be legal circuit-breakering/"mixture of 240V and 120V"-wise on a 4-wire
circuit, you'd have to put in a breaker for the 120V tap, and treat the circuit
as a subfeed. Only legal if it's four wire. May _not_ be sufficient to
avoid the "do not mix" provisions. See (4) too.

By the time you buy the breaker, housing, and cut up the walls, it'll
likely cost just as much if not more than running a new 120V feed or
tapping off something else for the washer.

4) Modern separate dryer/washers are likely to draw considerably more
power when operated simultaneously than a combo washer/dryer (even a
stacked) unit.

Adding the washer to the dryer circuit is likely to break the 80% ampacity
code rule, and perhaps even trip the breaker. Ie: 4800W dryer element
plus a few amps @ 120V for the dryer motor, then add a largish washer
motor on one of the 240V legs - especially if you didn't trace the
appliance wiring and arranged the washer motor to be on the opposite
leg from the dryer motor.

Best case you only violate the 80% code rule. Good chance of tripping
the breaker.
--
Chris Lewis, Una confibula non set est
It's not just anyone who gets a Starship Cruiser class named after them.