We should not make fun of others.
Can I also say “… gli altri.”
And if yes, how does it sound in comparison?
I am asking because a literal translation of “il prossimo” sounds very oldfashioned in German: “Liebe Deinen Nächsten …” = “Love thy neighbour …”
Ciao. “Don’t make fun of the next” meaning “others” might be a useful translation. “Gli altri” - the others, might not work so well in Italian. “Don’t make fun of the others” might lead us to ask “Which others?”
Over to a madrelingua 
Both “il prossimo” and “gli altri” work as a translation of “other people”. “Gli altri” feels more conversational, “il prossimo” feels a bit judgmental.
I don’t feel “il prossimo” is particularly old fashioned; although “ama il prossimo tuo…” indeed would be strange to use outside of a citation.
I think in the meaning of “people around you” you always use it with the determinative article, in the singular number.
It is a word that comes from the irregular Latin superlative “proximus/um/a”, menaning “the closest”.
So as an adjective it now means “the next”, and as a name “people close to/around you”.
Grazie come sempre! Molto utile.