Refering to the two verses below you can easily tell that we should abstain from insulting any one whether it was a non believer or a believer:
6:108 And do not insult those they invoke other than Allah , lest they
insult Allah in enmity without knowledge. Thus We have made pleasing
to every community their deeds. Then to their Lord is their return,
and He will inform them about what they used to do.
49:11 O you who have believed, let not a people ridicule [another]
people; perhaps they may be better than them; nor let women ridicule
[other] women; perhaps they may be better than them. And do not insult
one another and do not call each other by [offensive] nicknames.
Wretched is the name of disobedience after [one's] faith. And whoever
does not repent - then it is those who are the wrongdoers.
Even if someone verbally abuses us that still does not give us the right to abuse them as well as two wrongs don't make a right and we will be rewarded for our patience
"The Muslim who mixes with the people and bears patiently their
hurtful words, is better than one who does not mix with people and
does not show patience under their abuse." (Mishkat, Book: Ethics, ch.
'Gentleness, modesty and good behaviour')
(3:185) "You will certainly hear much abuse from the followers of
previous books and from the idol-worshipping people. And if you are
patient and keep your duty -- this is surely a matter of great
resolution."
This is one of my favourite hadeeths, it indicates that we should have more control over our speech and think of what we are about to say before saying it for it may harm someone:
The Messenger of Allah, Muhammed - SallAllahu 'alaihi wa sallam -
said, “Whoever believes in Allah and the Last Day should speak good or
keep silent.”
We are not limited to just abstaining from verbal insults rather from back biting others and speaking ill of them in their absence:
Narrated by Mughira bin Shu’ba - radiAllahu' anhu - : The Prophet -
SallAllahu 'alaihi wa sallam - said, “Allah has forbidden for you:(1)
to be undutiful to your mothers, (2) to bury your daughters alive, (3)
not to pay the rights of the poor and others (i.e. charity) and (4) to
beg of men. And Allah has hated for you: (1) Qil and Qal (sinful and
useless talk, like backbiting or that one talks too much about
others), (2) to ask too many questions (in disputed religious matters,
etc.), (3) to waste your wealth (by extravagance with lack of wisdom
and thinking).” - Sahih al-Bukhari [3:591]
Allah Knows best, inshalah this helped