Is it essential to wash plates after they have been used and washed by non-Muslims?

The basic principle concerning plates is that they are pure,
whether they are used by a Muslim, one of the People of the Book (a Jew or a
Christian) or anyone else, unless it is certain that they are naajis
(impure). Hence the majority of fuqaha’ are of the view that it is
permissible to use the vessels of the kaafirs. They quote a number of things
as evidence for that, including the following:  

1 – Allaah has permitted us to eat the food of the people of
the Book, i.e., meat slaughtered by them. It is known that it will sometimes
come to us cooked in their vessels, which indicates that it is permissible
to use their vessels. 

2 – The Prophet (peace and blessings of Allaah be upon
him) was invited to eat some barley bread and other food by a Jewish boy. 
Narrated by Ahmad, classed as saheeh by al-Albaani in Irwa’ al-Ghaleel,
1/71. 

3 – The Prophet
(peace and blessings of Allaah be upon him) and his companions did wudoo’
from the leather water skin of a mushrik woman. Narrated by al-Bukhaari,
337; Muslim, 682.  

This evidence indicates that it is permissible to use the
vessels of the kuffaar. 

But if we know that they cook pork or dead meat in their
vessels, or they drink wine from them, then it is better to avoid using them
unless it is necessary and we can find no alternative. Then we may wash them
and eat from them. If they wash them, then we do not have to wash them
again, and it is not necessary to wash them three times, rather they should
be washed until there are no traces of their food and drink left on them. 

The evidence for that is the report narrated by al-Bukhaari
(5478) and Muslim (3567) from Abu Tha’labah al-Khushani who said: I said, “O
Prophet of Allaah, I live in a land where there are some of the People of
the Book; can we eat from their vessels?” He said: “As for what you said
about the People of the Book, if you can find anything else, do not eat from
their vessels, but if you cannot find anything else then wash them and eat
from them.” 

***** 

This may be understood as referring to those among them who
use their vessels for haraam things, because of the report from Abu Dawood
(3839): “We live next to some of the People of the Book who cook pork in
their vessels and drink wine in their vessels.” The Messenger of Allaah
(peace and blessings of Allaah be upon him) said: “If you can find anything
else, eat from them and drink from them, but if you cannot find anything
else, then wash them with water and eat and drink.” Classed as saheeh by
al-Albaani in Saheeh Abi Dawood. 

Al-Khattaabi said: 

The basic principle in this matter is that if it is known
that the mushrikeen cook pork in their pots and drink wine from their
vessels, then it is not permissible to use them until after they have been
washed and cleaned.” 

End quote from ‘Awn al-Ma’bood. 

The phrase “If you can
find anything else, eat from them and drink from them” means eat and drink
from those other vessels. This command means that it is mustahabb, according
to the majority of fuqaha’, i.e., it is mustahabb to avoid these vessels,
and it is makrooh to use them even if they are washed, unless there are no
others available in which case it is no longer makrooh. 

Al-Nawawi (may Allaah have mercy on him) said in Sharh
Muslim (13/80): 

The reason why it is better not to eat from them even after
washing is because they are regularly used for naajis things.” End quote. 

Shaykh Ibn ‘Uthaymeen
(may Allaah have mercy on him) said in al-Sharh al-Mumti’, 1/68: 

The hadeeth of Abu
Tha’labah al-Khushani according to which the Messenger (peace and
blessings of Allaah be upon him) said: “Do not eat from them unless you
cannot find anything else, in which case wash them and eat from them”
indicates that it is better not to use them, but many of the scholars
interpreted this hadeeth as referring to people who are known to handle
naajis things, such as eating pork and so on. They said: The Prophet
(peace and blessings of Allaah be upon him) forbade eating from their
vessels unless we cannot find anything else, in which case we should wash
them and eat from them. This is a good interpretation, and is implied by the
basic principles of sharee’ah. End quote. 

To sum up: 

If these people do not use those vessels for drinking wine or
eating pork or dead meat, then it is permissible to use them. If they do use
them for haraam or naajis kinds of food and drink, then it is better not to
use them if you can find something else. If you cannot find something else
then you can use them after they have been washed, whether it is you or they
who wash them. 

And Allaah knows best.