Do Arahants have Vinnana and Namarupa?

Do you have the wind?

If you have, then you can control it. Then you can stop the wind, give it to others and so on.

If you haven't the wind, then it can't blow on you, because you have no wind.

Arahants who are still living are those with "fuel" remaining and sense faculties (salayatana) still operating. The "fuel" here refers to the five aggregates. Hence Vinnana, one of the five aggregates, is still existing. The five aggregates have stopped burning with the fires of passion, aversion and delusion, but they are like glowing embers. When the arahant passes away, then the "fuel" is no more remaining.

Then, what about namarupa? It is also still existing for living arahants. Contact (phassa) and feeling (vedana) also still exists for living arahants. However, craving (tanha) has ended for arahants. Without craving, there is no clinging (upadana). Without clinging, there is no becoming (bhava) in the future i.e. no suffering here or anywhere else after death. Arahants are also free of ignorance (avijja).

According to Itivuttaka 44:

This was said by the Blessed One, said by the Arahant, so I have heard: "Monks, there are these two forms of the Unbinding property. Which two? The Unbinding property with fuel remaining, & the Unbinding property with no fuel remaining.

And what is the Unbinding property with fuel remaining? There is the case where a monk is an arahant whose fermentations have ended, who has reached fulfillment, finished the task, laid down the burden, attained the true goal, ended the fetter of becoming, and is released through right gnosis. His five sense faculties still remain and, owing to their being intact, he is cognizant of the agreeable & the disagreeable, and is sensitive to pleasure & pain. His ending of passion, aversion, & delusion is termed the Unbinding property with fuel remaining. (Note1)

And what is the Unbinding property with no fuel remaining? There is the case where a monk is an arahant whose fermentations have ended, who has reached fulfillment, finished the task, laid down the burden, attained the true goal, ended the fetter of becoming, and is released through right gnosis. For him, all that is sensed, being unrelished, will grow cold right here. This is termed the Unbinding property with no fuel remaining." (Note2)

Note1 and Note2 (commentary by Thanissaro Bhikkhu):
With fuel remaining (sa-upadisesa) and with no fuel remaining (anupadisesa): The analogy here is to a fire. In the first case, the flames are out, but the embers are still glowing. In the second, the fire is so thoroughly out that the embers have grown cold. The "fuel" here is the five aggregates. While the arahant is still alive, he/she still experiences the five aggregates, but they do not burn with the fires of passion, aversion, or delusion. When the arahant passes away, there is no longer any experience of aggregates here or anywhere else.

First of all, it is incorrect to say "arhats have vijnana (etc)". The word "have" implies someone who has, but in case of an arhat there's no entity, no one to "have" vijnana. The right way to phrase this question is to say: In case of an arhat, are there vijnana and namarupa?"

Second, an arhat will not be reborn exactly because there is no entity to be reborn, and this realization is part of cessation of ignorance which normally keeps the Twelve Nidanas rolling.

Third, in this context vijnana and namarupa refer to nidanas, not to skandhas. The twelve nidanas describe the process of gradual taking shape and hardening of a certain subjective world and the person living in it. Vijnana and namarupa refer to early stages of this process. First, there develops a tendency to recognize and discriminate, and from that tendency appear various objects and entities. That's what vijnana and namarupa refer to in context of twelve nidanas. They refer to the naive, ignorant discrimination and to the illusory entities that it delineates.

In case of an arhat, the discernment is no longer ignorant and it therefore does not posit illusory entities. Buddha clearly said, with cessation of ignorance, there's cessation of tendencies, with cessation of tendencies there's cessation of vijnana etc. This refers to cessation of naive, ignorant automatic discrimination which contrives objects and then believes they are absolutely ontologically real.

So we are not saying that the skandhas disappear with attainment of arhathood, not at all. Arhathood is cessation of nidanas, the causal chain that leads to emergence and hardening of the illusory world and the illusory self.

And of course arhats are not mindless dummies. In case of an arhat there's a function of right discernment. This function is called prajna. The object of this discernment is yatabhuta, things as they truly are. This refers first and foremost to seeing samskaras as samskaras, instead of the fixed entities. Because samskaras are seen as samskaras, they are seen as Anicca, Dukkha, Anatta, and at Mahayana level, as Shunya.

So the short answer is, no, in case of an arhat there's no vijnana and namarupa - there's prajna and samskara/shunyata.