Many people in the world celebrate and play pranks on each other on April 1.
Is it halachically permissible to play pranks on people or in any kind celebrate April Fool's Day?
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I think it'll be a bit dfficult to give overall rules about whether 'pranks' are allowed, since what one might consider a 'prank' is quite varied, with some items being forbidden and some being permitted.
With that said, there are still certain things that are forbidden. Once we list all the 'pranks' that are forbidden, we can figure out what is permitted by process of elimination. :)
One category of forbidden 'pranks' is anything that involves stealing an item, even if there is no monetary damage. This is brought down in Kitzur Shulchn Aruch 182:3:
אֲפִלּוּ לִגְנוֹב עַל דַּעַת לְהַחֲזִיר, אֶלָּא שֶׁרוֹצֶה לְצַעֲרוֹ קְצָת אוֹ בְּדֶרֶךְ שְׂחוֹק, גַּם כֵּן אָסוּר.
Even stealing with the intention of returning it, but just for the sake of annoying someone, or to tease someone, is also forbidden.
Note that the above is when there is no damage to the item. If there is actual damage involved, it would clearly be more severe.
There may also be issues of humiliating a person (see Kitzur Shulchan Aruch Chapter 63, which while mostly dealing with business-related deceiving, has several non-business applications) and lying (see for example Kitzur Shulchan Aruch 181:7) but those were mentioned in @AlBerko's answer, so I won't focus on them here.
Fooling others [Jews] by saddening them even minutely is prohibited Deorayso even if they forgive afterward:
כשם שאונאה במקח וממכר כך אונאה בדברים:
לא יאמר לו בכמה חפץ זה והוא אינו רוצה ליקח,
אם היה בעל תשובה לא יאמר לו זכור מעשיך הראשונים,
אם הוא בן גרים לא יאמר לו זכור מעשה אבותיך
שנאמר (שמות כב, כ) וגר לא תונה ולא תלחצנו
Just as there is a prohibition against exploitation [ona’a] in buying and selling, so is there ona’a in statements, i.e., verbal mistreatment. The Mishna proceeds to cite examples of verbal mistreatment. ...
See also Shu"A Cho"M 228:
"אסור להונות בדברים ולגנוב דעת הבריות ולרמות במקח וממכר"
I would definitely add the prohibition of lying (מדבר שקר תרחק), which is less stringent but still important.