Can a Warforged suffer from magical exhaustion?

The Warforged Resilience trait (WGtE, p. 68) states:

You were created to have remarkable fortitude, represented by the following benefits.

• You have advantage on saving throws against being poisoned, and you
  have resistance to poison damage.


• You are immune to disease.


• You don’t need to eat, drink, or breathe.


• You don’t need to sleep and don’t suffer the effects of exhaustion
  due to lack of rest, and magic can’t put you to sleep.
  

Is the Warforged just immune to exhaustion due to not taking a long rest? Or can magic cause it to receive levels of exhaustion, even though it cannot put it to sleep?

They can suffer exhaustion, just not due to lack of rest.

"Exhaustion due to lack of rest" is carefully qualified. There are creatures that are immune to "exhaustion" (such as the Ghost). This is different. Lack of rest is just one of many ways to become exhausted; there's also starvation (though Warforged are immune to that too), exposure to extreme weather, forced marching, etc. Magic that directly inflicts exhaustion is separate from any of those.

(Forced marching is a gray area, in that it's caused by traveling for more than 8 hours in a day. At a normal travel pace, you travel 8 hours and then stop and rest, so the cause of your exhaustion is that you're continuing to walk instead of resting. Is that "exhaustion due to lack of rest"? I'd likely rule that it is, just because marching hundreds of miles without stopping seems like exactly the kind of thing Warforged should be good at.)

The protection from being magically forced to sleep is mechanically separate but conceptually related. What this paragraph is telling us is that Warforged physiology doesn't include a need for sleep. This means they don't suffer harm from not sleeping, but it also means that magic that reaches into people and hotwires the "time to sleep now" circuit will be ineffective, because they don't have that circuit.

Yes, they can be exhausted by magic

You don’t need to sleep and don’t suffer the effects of exhaustion due to lack of rest, and magic can’t put you to sleep. (WgtE)

Magic-induced exhaustion such as the one induced by sickening radiance has nothing to do with lack of rest, it is simply a magic effect.

When a creature moves into the spell’s area for the first time on a turn or starts its turn there, that creature must succeed on a Constitution saving throw or take 4d10 radiant damage, and it suffers one level of exhaustion and emits a dim, greenish light in a 5-foot radius. (XGE 164)

Such effects are not included in the warforged's resilience feature, only effects directly caused by lack of rest. In fact, (as Mark Wells points out in their answer) there are even other non-magical ways for exhaustion to be given (starvation, hot or cold temperatures, forced marching, etc) and warforged will also be susceptible to those.

If a warforged were intended to have a blanket resistance to exhaustion, they would not have included the words "due to lack of rest".